GIFT  OF 

V;illiamaMuart   Smith 


ENGINEERING  LIBRARY 


• 


THE 


RELATIVE  PROPORTIONS 


OF   THE 


STEAM-ENGINE: 


BEING 


A  RATIONAL  AND  PRACTICAL  DISCUSSION  OF  THE  DIMEN- 
SIONS OF  EVERY  DETAIL  OF  THE  STEAM-ENGINE. 


BY 

WILLIAM  DENNIS  MARKS, 

\i 

WHITNEY  PROFESSOR  OF  DYNAMICAL  ENGINEERING  IN  THK 
UNIVERSITY  OF  PENNSYLVANIA. 


WITH  NUMEROUS  DIAGRAMS. 


SECOND  EDITION.     REVISED  AND  ENLARGED. 


J.  B.  LIPPINCOTT    &    CO. 

1884 


Entered  according  to  Act  of  Congress,  in  the  year  1878,  by 

J.  B.  LIPPINCOTT   &   CO., 
In  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


COPYRIGHT,  1883,  BY  J.  B.  LIPPINCOTT  &  Co. 


GIFT  OF 


-v. 

ENGINEERING  LIBRARY 


'     PMuduphiu. 


PREFACE. 


IT  is  a  source  of  regret  to  the  author  of  these  Lectures 
that  none  of  the  distinguished  writers  upon  Mechanics  or 
the  Steam-engine  have  undertaken  to  give,  in  a  simple  and 
practical  form,  rules  and  formulae  for  the  determination  of 
the  relative  proportions  of  the  component  parts  of  the 
steam-engine. 

The  authors  of  the  few  works  as  yet  published  in  the 
English  language  either  entirely  ignore  the  proportions  of 
the  steam-engine  or  content  themselves  with  scanty  and 
general  rules — Rankine  excepted,  who  in  the  attempt  to  be 
brief  is  sometimes  obscure,  leaving  many  gaps  in  the  im- 
mense field  which  he  has  attempted  to  cover.  This  defi- 
ciency in  the  literature  of  the  steam-engine  is  remarkable, 
because  the  problem  which  the  mechanical  engineer  is  most 
frequently  called  upon  to  solve  is  the  determination  of  the 
dimensions  of  its  various  parts. 

From  time  to  time  hand-books  of  the  steam-engine  have 
been  published  giving  practical  (?)  rules,  the  result  of  obser- 
vation of  successful  construction ;  and  with  these  rules  the 
practising  engineer,  who  has  little  time  for  original  investi- 
gation, has  had  to  content  himself.  It  is  of  course  reasonable 
to  limit  the  correctness  of  these  rules  to  cases  in  which  all 

869476 


4  PREFACE. 

the  conditions  are  the  same,  as  in  the  case  or  cases  from 
which  these  rules  have  been  derived,  thus  placing  a  serious 
obstruction  in  the  way  of  improvement  or  alteration  of  de- 
sign, and  rendering  the  rules  worse  than  useless — even  dan- 
gerous in  many  cases. 

"  The  usual  resource  of  the  merely  practical  man  is  pre- 
cedent, but  the  true  way  of  benefiting  by  the  experience  of 
others  is  not  by  blindly  following  their  practice,  but  fry 
avoiding  their  errors,  as  well  as  extending  and  improving 
what  time  and  experience  have  proved  successful.  If  one 
were  asked,  What  is  the  difference  between  an  engineer  and 
a  mere  craftsman  ?  he  would  well  reply  that  the  one  merely 
executes  mechanically  the  designs  of  others,  or  copies  some- 
thing which  has  been  done  before,  without  introducing  any 
new  application  of  scientific  principles,  while  the  other 
moulds  matter  into  new  forms  suited  for  the  special  object 
to  be  attained,  and  lets  his  experience  be  guided  and  aided 
by  theoretic  knowledge,  so  as  to  arrange  and  proportion 
the  various  parts  to  the  exact  duty  they  are  intended  to 

fulfil. 

"'For  this  is  art's  true  indication, 

When  skill  is  minister  to  thought, 
When  types  that  are  the  mind's  creation 
The  hand  to  perfect  form  hath  wrought.' " 

STONEY'S  Theory  of  Strains. 

Zeuner,  in  his  elegant  Treatise  on  Valve  Gears,  transla- 
ted by  M.  Miiller,  has  laid  the  foundation  for  the  treatment 
of  slide-valve  motions  for  all  time,  and  in  his  Mechanisehe 
Wdrmetheorie  has  carried  the  application  of  the  mechan- 


PREFACE.  5 

ical  theory  of  heat  to  the  steam-engine  as  far  as  the  present 
state  of  the  science  of  Thermo-dynamics  will  permit. 

Poncelet,  in  his  Mecanique  appliquee  aux  Machines,  has 
most  thoroughly  treated  some  members  of  the  steam-engine, 
neglecting  others  of  as  great  practical  importance. 

Him,  in  his  Theorie  mecanique  de  la  Chaleur,  gives 
us,  besides  a  very  able  treatise  on  the  science  of  Thermo- 
dynamics, a  valuable  series  of  experiments  upon  the  steam- 
engine  itself,  confirming  Joule's  results. 

A  translation  of  Der  Constructeur,  by  F.  Reuleaux, 
would,  if  made,  add  much  to  our  knowledge  of  the  proper 
proportions  of  the  steam-engine,  as  well  as  of  other  ma- 
chines. 

A  rational  and  practical  method  of  determining  the 
proper  relative  proportions  of  the  steam-engine  seems  as 
yet  to  be  a  desideratum  in  the  English  literature  of  the 
steam-engine;  and  these  Lectures  have  been  written  with 
that  feeling,  purposely  omitting  the  consideration  of  such 
topics  as  have  already  in  many  cases  been  over-written,  and 
considering  only  those  which  have  not  received  the  atten- 
tion which  their  importance  demands. 

In  the  choice  of  a  factor  of  safety — a  matter  wherein 
opinions  widely  differ — the  author,  guided  by  considerations 
set  forth  in  Weyrauch's  Structures  of  Iron  and  Steel,  trans- 
lated by  DuBois,  has  fixed  upon  10  as  being  the  most  cor- 
rect value.  If  any  of  our  readers  should  prefer  a  different 
factor,  the  formulae  deduced  will  be  correct  if  the  actual 
steam-pressure  per  square  inch  is  divided  by  10  and  multi- 
i* 


6  PREFACE. 

plied  by  the  preferred  factor  of  safety,  and  the  result  used 
in  the  place  of  the  actual  steam-pressure. 

In  reducing  all  the  required  dimensions  of  parts  of  the 
steam-engine  to  functions  of  the  boiler-pressure  or  mean 
steam-pressure  in  the  cylinder  per  square  inch,  of  the  diam- 
eter of  the  steam-cylinder,  length  of  stroke,  number  of 
strokes  per  minute,  and  horse-power,  he  trusts  that  he  has 
put  the  formulae  in  the  simplest  possible  form  for  immediate 
use. 

It  is  indeed  in  this  transformation  of  the  formulae  for  the 
strength  of  materials  that  the  usefulness  of  the  book  lies ; 
for  the  practitioner,  once  satisfied  of  their  correctness,  has 
but  to  insert  quantities  fixed  at  the  commencement  of  his 
design,  and  derive  from  the  formulae  the  required  dimen- 
sions, being  relieved  of  many  formulse  and  details  connected 
with  the  applications  of  statics  to  the  strength  and  elasticity 
of  materials. 

The  constant  references  to  the  fourth  section  of  Weis- 
bach's  Mechanics  of  Engineering  are  necessary,  as  it  is 
no  part  of  the  author's  plan  to  discuss  the  strength  and 
elasticity  of  materials  any  further  than  it  is  necessary  to  do 
so  in  their  application  to  the  steam-engine.  Those  unac- 
quainted with  this  branch  of  mechanical  engineering  will 
nowhere  find  it  treated  with  greater  simplicity  and  thor- 
oughness. Other  references  have  been  made  for  the  pur- 
pose of  directing  the  reader  to  such  sources  as  have  been 
drawn  upon  in  the  consideration  of  topics  discussed  in  this 
work. 


PREFACE.  1 

"We  who  write  at  this  late  day  are  all  too  much  in- 
debted to  our  predecessors,  whether  we  know  it  or  not,  to 
complain  of  those  who  borrow  from  us ;"  and  each  of  us  is 
only  able  to  make  his  relay,  taking  up  his  work  where 
others  have  left  it. 

The  lack  of  accurate  experimental  data  has,  in  many 
cases,  forced  the  writer  to  make,  perhaps,  bold  assumptions 
which  may  not  prove  entirely  correct ;  however,  as  in  all 
cases  the  method  of  reasoning  is  given,  the  reader,  where 
he  is  in  possession  of  more  accurate  data,  can  modify  by 
substitution. 

The  accidental  loss  of  all  of  the  original  manuscript  and 
drawings  of  these  Lectures,  and  the  necessity  of  rapidly  re- 
writing them  for  use  in  daily  instruction,  have  caused  the 
work  to  be  more  abbreviated  than  was  originally  intended. 

Deeply  sensible  of  the  many  unavoidable  deficiencies  of 
this  little  work,  even  in  the  limited  field  covered,  its  author 
still  hopes  that  it  will  aid  in  the  diffusion  and  advancement 
of  real  knowledge,  upon  whose  progress  the  prosperity  of 
our  civilization  depends. 

W.  D.  M 

UNIVERSITY  OF  PENNSYLVANIA, 
Philadelphia,  1878.      - 


PREFACE  TO  THE  SECOND  EDITION. 


THE  kind  reception  of  the  First  Edition  of  this  work 
has  afforded  the  author  an  unexpected  pleasure,  for  which 
he  is  very  grateful. 

The  pleasure  is  the  greater  because  it  has  come  from  an 
unlooked-for  direction.  Those  actively  engaged  in  design- 
ing engines  rarely  turn  to  books  for  instruction,  and  are 
quite  frequently  heard  to  complain  that  they  find  in  them 
a  good  deal  that  they  do  not  care  to  know,  and  nothing 
that  they  can  utilize.  Among  the  added  lectures  the 
author  wishes  particularly  to  call  the  attention  of  readers 
to  The  Cheapest  Point  of  Gut -Off  and  to  The  Errors  of  the 
Zeuner  Diagram.  The  former  will  prove  of  interest  to  all, 
whether  designers  or  users.  The  latter  is  not  written  with 
a  desire  to  detract  from  the  honor  due  to  Prof.  Zeuner,  but 
rather  to  perfect  the  great  discovery  of  the  law  of  valve- 
motions  due  to  his  genius. 

W.  D.  M. 

UNIVERSITY  OP  PENNSYLVANIA,  1883. 


CONTENTS. 


ART.  CHAPTEE  I.  PACK 

1.  INTRODUCTORY 13 

2.  The  Steam-Cylinder 13 

3.  Indicated  Horse-Power 15 

4.  Thickness  of  the  Steam-Cylinder 17 

5.  Thickness  of  the  Cylinder-Heads 19 

6.  Cylinder-Head  Bolts 20 

CHAPTEK  II. 

7.  Standard  Screw-Threads  for  Bolts 22 

8.  The  Steam-Chest 25 

9.  The  Steam-Ports..... 28 

10.  The  Piston-Head 29 

11.  The  Piston-Eod 29 

12.  Wrought-Iron  Piston-Kod 30 

13.  Steel  Piston-Kod 34 

CHAPTER  III. 

14.  Comparison   and   Discussion   of   Wrought-Iron   and  Steel 

Piston-Rods 37 

15.  Keys  and  Gibs 37 

16.  Wrought-Iron  Keys 40 

17.  Steel  Keys 43 

18.  The  Cross-Head..... 44 

19.  Area  of  the  Slides...,                                                              .  46 


10  CONTENTS. 

ART  CHAPTER  IV. 

20.  Stress  on  and  Dimensions  of  the  Guides 49 

21.  Distance  between  Guides 51 

22.  The  Connecting-Rod 53 

23.  Wrought-Iron  Connecting-Rod 55 

CHAPTER  V. 

24.  Steel  Connecting-Rod 59 

25.  General  Remarks  concerning  Connecting-Rods 62 

26.  Connecting-Rod  Straps 63 

27..  Wrought-Iron  Straps 63 

28.  Steel  Strap , 65 

CHAPTER  VI. 

29.  The  Crank-Pin  and  Boxes 66 

30.  The  Length  of  Crank-Pins 70 

31.  Locomotive  Crank-Pins,  Length  and  Diameter  of 73 

CHAPTER  VII. 

32.  Diameter  of  a  Wrought-Iron  Crank-Pin  for  a  Single  Crank.  74 

33.  Steel  Crank-Pins 75 

34.  Diameters  of  Crank-Pins  from  a  Consideration  of  the  Pres- 

sure upon  them 76 

35.  Of  the  Action  of  the  Weight  and  Velocity  of  the  Recipro- 

cating Parts 76 

CHAPTER  VEIL 

36.  The  Single  Crank 85 

37.  Wrought-Iron  Single  Crank 90 

CHAPTER  IX. 

38.  Steel  Single  Crank 93 

39.  Cast-Iron  Cranks 95 

40.  Keys  for  Shafts 95 

41.  Wrought-Iron  Keys  for  Shafts 98 


CONTENTS.  11 

ART.  PAGB 

42.  Steel  Keys  for  Shafts 99 

43.  The  Crank  or  Main  Shaft 99 

44.  Shaft  Subjected  to  Flexure  only 100 

45.  Wronght-Iron  Shaft,  Flexure  only 101 

CHAPTEK  X. 

46.  Steel  Shaft,  Flexure  only 103 

47.  Shaft  Subjected  to  Torsion  only 103 

48.  Wrought-Iron  Shaft,  Torsion  only 104 

49.  Steel  Shaft,  Torsion  only 104 

50.  Shaft  Submitted  to  Combined  Torsion  and  Flexure 105 

51.  Flexure  and  Twisting  of  Shafts 108 

52.  Comparison  of  Wrought-Iron  and  Steel  Crank-Shafts Ill 

53.  Journal-Bearings  of  the  Crank-Shaft Ill 


CHAPTEK  XI. 

54.  Double  Cranks 115 

55.  Triple  Cranks 118 

56.  The  Fly- Wheel 120 

57.  Fly-Wheel,  Single  Crank 123 

58.  Fly- Wheel,  Double  Crank,  Angle  90° 127 


CHAPTEE  XII. 

59.  Fly- Wheel,  Triple  Cranks,  Angles  120° 129 

60.  Of  the  Influence  of  the  Point  of  Cut-Off  and  the  Length  of 

the  Connecting-Hod  upon  the  Fly- Wheel 131 

CHAPTEE  XIII. 

61.  The  Weight  of  the  Eim  of  Fly- Wheels 137 

62.  Value  of  the  Coefficient  of  Steadiness 138 

3.  Area  of  the  Cross-Section  of  the  Eim  of  a  Fly- Wheel 139 

64.  Balancing  the  Fly- Wheel 140 

65.  Speed  of  the  Eim  of  the  Fly-Wheel 142 


12  CONTENTS. 

ART.  CHAPTER  XIV.  PAGK 

66.  Centrifugal  Stress  on  the  Arms  of  a  Fly- Wheel 143 

67.  Tangential  Stress  on  the  Arms  of  a  Fly- Wheel  for  a  Single 

Crank 147 

68.  Work  Stored  in  the  Arms  of  a  Fly- Wheel 150 

69.  The  Working-Beam 151 

70.  General  Considerations 151 

71.  Note  on  the  Taper  of  Connecting-Rods 154 

CHAPTER  XV. 

72.  The  Limitations  of  the  Steam-Engine 161 

CHAPTER  XVI. 

73.  The  Cheapest  Point  of  Cut-Off. 176 

CHAPTER  XVH. 

74.  The  Errors  of  the  Zeuner  Valve-Diagram 187 

TABLES. 

75.  TABLE  V.— The  Elasticity  and  Strength  of  Extension  and 

Compression 212 

76.  TABLE  VI.— The   Elasticity  and  Strength  of  Flezure   or 

Bending 213 

77.  TABLE  VII.— The  Elasticity  and  Strength  of  Torsion 214 

78.  TABLE  VIII.— The  Proof-Strength  of  Long  Columns 215 

NOTE.— These  Tables  will  be  found  to  contain  almost  all  the  formula 
referred  to  in  these  Lectures. 


.-, ' •  •  •  •         -'  *  ,> 
THEJ 


RELATIVE  PBOPOETIONS 

OF  THE 

STEAM-ENGINE. 


CHAPTER    I. 

(1.)  Introductory. — In  the  present  work,  unless  spe- 
cially stated,  the  single-cylinder,  double-acting  steam-engine 
only  will  be  the  subject  of  discussion. 

In  making  use  of  the  following  rules  and  formulae,  if  a 
non-condensing  engine  is  under  consideration,  the  pressures 
per  square  inch  above  the  atmosphere,  or  as  registered  by 
an  ordinary  steam-gauge,  must  be  used.  If  a  condensing 
engine  be  considered,  fifteen  pounds  per  square  inch  must  be 
added  to  the  pressures  above  the  atmosphere. 

While  it  is  impossible  to  take  up  every  form  of  steam- 
engine  which  has  been  invented,  the  formulae  are  sufficiently 
general  to  admit  of  adaptation  to  any  form  of  engine  which 
the  engineer  may  wish  to  devise. 

(2.)  The  Steam-Cylinder. — In  many  cases  occurring 
in  practical  Mechanics  other  considerations  than  economy 
of  steam  determine  either  the  stroke  or  the  diameter  of  the 
steam-cylinder.  When,  however,  these  dimensions  are  not 
fixed  by  other  considerations,  that  of  economy  of  steam 
should  have  the  precedence,  as  being  a  constant  source  of 
gain ;  and  it  being  demonstrable  by  the  differential  calculus 
that  the  surface  of  any  cylinder  closed  at  the  ends  and  en- 
closing a  given  volume  is  a  minimum  when  the  diameter 
2  13 


14 


THE  RELATIVE  PROPORTIONS 


of  that  cylinder/ "is  equfeLto  jits;  length,*  it  follows  that  any 
given  volume  *oi" 'steam  *ha§  a'  inittimum  of  surface  of  con- 
densation /ftnji'jJon^QcJuSnS^j  loses; Jess  ;by  condensation  than 
it  would  in  any  'other  'cylmcfef  of 'equal  volume  and  differ- 
ent relative  dimensions ;  and  therefore  that  the  best  relative 
proportions  of  the  stroke  and  diameter  of  the  steam-cylinder 
are  attained  when  they  are  equal. 

The  importance  of  the  action  of  the  walls  of  the  steam- 
cylinder  in  condensing  steam,  and  the  inability  of  a  steam- 
jacket  to  do  more  than  keep  the  cylinder  warm,  without 
actually  communicating  any  appreciable  amount  of  heat  to 
the  enclosed  steam,  are  becoming  more  clearly  recognized 
among  engineers,  and  are  forcing  them  to  adopt  higher  pis- 
ton-speeds, "f  to  take  means  of  reducing  the  surface  of  con- 

*  Demonstration. — Let  the  surface  of  the  cylinder  =  S  —  nyx + -~ .  (1) 

"    "  volume   "  "       =F=-^.          (2) 

47 
From  eq.  (2)  we  have  7rx  =  — r  or  ^xy  =  4.Vy~l. 

y 

Fio.  I-  Substituting  in  eq.  (1),  we  have 


Differentiating,  we  have 

xL-4Fjr2  +  7ry  =  i 


giving 


4V 

Substituting  this  value  of  y  in  eq.  (2),  we  have  also  z=\/-^r->  show- 
ing, since  the  second  diff.  coef.  is  positive,  a  minimum. 

f  For  a  full  discussion  of  the  consumption  of  steam,  see  A  Treatise 
on  Steam,  chap,  iv.,  Graham ;  "  Experiments  with  a  Steam-Engine," 
translated  from  Bulletin  de  la  Societe  Industriel  de  Mulhouse,  in  Journal 
of  the  Association  of  Entfg  Societies,  May,  J.883,  Hirn-Smith;  also 
Chapters  XV.  and  XVI. 


OF  THE  STEAM-ENGINE.  15 

densation  in  the  cylinder,  and  to  superheat  the  steam  to  a 
limited  extent  before  its  introduction  into  the  engine. 

If  the  cylinder  and  heads  be  of  a  uniform  thickness,  the 
quantity  of  metal  required  to  form  a  cylinder  of  the  de- 
sired volume  is  nearly  a  minimum  :  we  say  "nearly,"  because 
sufficient  length  must  be  added  to  the  cylinder  to  provide 
for  the  thickness  of  the  piston-head  and  the  required  clear- 
ance at  the  ends. 

The  use  of  shorter  cylinders  than  has  hitherto  been  cus- 
tomary has  the  advantage  of  reducing  the  piston-speed,  and 
consequently  the  wear  upon  the  piston-packing,  for  any 
given  number  of  revolutions,  although  the  wear  upon  the 
interior  of  the  steam-cylinder  is  constant  for  any  constant 
number  of  strokes  per  minute. 

(3.)  Indicated  Horse-Power.  —  In  the  present  work 
the  indicated  horse-power  only  will  be  referred  to  or  made 
use  of. 

Let  (HP)  =  the  indicated  horse-power. 

P  =the  mean  pressure  of  steam  on  the  piston- 

head  in  pounds  per  square  inch. 
L   =  the  length  of  stroke  in  feet. 
A  =  the  area  of  the  piston-head  in  square  inches. 
JV  =  the  number  of  strokes  (  =  twice  the  number 

of  revolutions  of  the  crank)  per  minute. 
We  have  the  well-known  formula, 

If  in  formula  (1)  we  make  the  length  of  stroke  in  inches 
equal  to  the  diameter  of  the  steam-cylinder,  it  becomes, 
letting  d  =  the  diameter  of  cylinder  in  inches, 


12x33000  ' 


16        THE  RELATIVE  PROPORTIONS 

and  reducing,  we  have  for  the  common  diameter  and  stroke 
of  a  steam-cylinder  of  any  assumed  horse-power, 

d-  79.59  -J^p.  (2) 

This  formula  gives,  for  any  assumed  horse-power,  mean 
pressure,  and  number  of  strokes  per  minute,  the  common 
diameter  and  stroke  of  the  steam-cylinder  in  inches. 

The  reduction  in  size,  and  the  consequent  economy  in 
using  steam,  resulting  from  assuming  the  pressure  per 
square  inch,  P,  and  the  number  of  strokes  per  minute,  N, 
as  large  as  circumstances  will  permit,  in  designing  an  engine 
of  any  desired  horse-power  (-HP),  will  at  once  be  perceived 
upon  inspection  of  formula  (2).  The  advantages  resulting 
from  high  pressures,  early  cut-off,  and  rapid  piston-speed 
will  be  more  thoroughly  discussed  in  Art.  35  of  this  work 

Example.  —  In  a  given  cylinder, 
Let  L  =  4  ft.  =  48  inches. 
"     d  =  32  inches. 
"    P  =  40  Ibs.  per  square  inch. 
"    N=  40  per  minute  =  20  revolutions  of  the  crank. 
Using  formula  (1),  we  have 

PLAN     40x4x804.25x40 

=156>  approx- 


3000 

If  now  we  assume  the  horse-power  (-HP)  =  156,  and 
Let,  as  before,         N=  40  per  minute, 

P=40  Ibs.  per  square  inch, 
we  have,  using  formula  (2), 

d  =  79.59  V^P  -  79.59  \l-^-  =  36.73  inches, 
*  PN  v  40x40 

for  the  required  common  stroke  and  diameter  of  a  cylinder 
of  equal  power  with  the  first. 


OF   THE  STEAM-ENGINE.  17 

(4.)  Thickness  of  the  Steam-Cylinder. — Steam-cylin- 
ders are  usually  made  of  cast  iron ;  and  in  order  that  the 
engine  may  be  durable,  this  casting  should  be  made  of  as 
hard  iron  as  will  admit  of  working  in  the  shop.  Steel  lining- 
cylinders  for  ordinary  cast-iron  cylinders  have  sometimes 
been  used,  and  have  well  repaid  in  durability  their  greater 
cost. 

Large  steam-cylinders  should  always  be  bored  in  either  a 
horizontal  or  vertical  position,  similar  to  that  in  which  they 
are  to  be  placed  when  in  use. 

Weisbach,  in  Art.  443,  Vol.  ii.,  of  the  Mechanics  of  En- 
gineering, gives  the  following  formula  for  the  thickness  of 
steam-cylinders : 

Let     t  =  the  thickness  in  inches  of  cast-iron  cylinder- walls. 
"    Pb  =  the  boiler-pressure  in  pounds  per  square  inch. 
"      d  =  the  diameter  of  the  cylinder  in  inches. 
Then 

I  =  0.00033P6d  +  0.8  inch,  (3) 

which  makes  0.8  inch  the  least  possible  thickness  of  a  steam- 
cylinder. 

Van  Buren,  in  Strength  of  Iron  Parts  of  Steam-Machinery, 
page  58,  establishes  the  following  formula  by  means  of  a 
discussion  of  a  72-inch  English  steam- cylinder  which  had 
been  found  to  work  well : 


V/2D 
• 


Reuleaux,  in  Der  Constructeur,  page  561,  gives  the  fol- 
lowing empirical  formula  for  the  completed  thickness  of 
steam-cylinders : 

<  =  0.8  inch  +  ~^.  (5) 


18  THE  RELATIVE  PROPORTIONS 

Inspection  of  these  differing  formulae,  all  founded  upon 
successful  practice,  would  lead  to  the  conclusion  that  it  is 
best  first  to  calculate  the  thickness  necessary  to  withstand 
the  pressure  of  the  steam,  and  then  to  make  an  addendum 
sufficient  to  provide  for  boring*  and  re-boring,  and  also  to 
give  the  cylinder  perfect  rigidity  in  position  and  form. 

Good  cast  iron  has  an  average  tensile  strength  of  18,000 
pounds  per  square  inch  cross-  section,  and  with  a  factor  of 
safety  of  10  gives  1800  pounds  per  square  inch  as  a  safe 
strain.  That  this  factor  is  not  too  large  will  be  conceded 
when  we  consider  that  with  some  forms  of  valve-motion  the 
admission  of  steam  to  the  cylinder  partakes  of  the  nature  of 
a  veritable  explosion. 

From  Weisbach's  Mechanics  of  Engineering,  Vol.  i.,  Sec. 
vi.,  Art.  363,  we  have,  if  we  take  safe  strain  =  1800  pounds, 


0.00028^.  (6) 

Example.  —  For  a  locomotive  cylinder, 

Let  Pb  =  150  pounds  per  square  inch. 
"      d  =  20  inches. 

From  formula  (3)  we  have  t  =  1.8    inches. 
"       (4)   "      "    2  =  1.34      " 

(5)  "      "     4-1.00      'l 

(6)  "      "     t=   .83      " 

Any  of  the  thicknesses  given  would  probably  serve  success- 
fully, and  about  U  inches  is  the  best  practice.     It  is  not 

*  The  best  means  of  securing  an  approximately  true  cylinder  is  to 
finish  to  size  with  a  shallow  broad  cut,  giving  a  rapid  feed  to  the  lathe 
or  boring-mill  tool.  If  the  cylinder  is  bored  with  a  fine  feed  and  deep 
cutting-tool,  the  gradual  heating  and  subsequent  cooling  are  apt  to 
make  the  interior  tapering  in  form,  as  well  as  to  require  the  running 
of  the  shop  out  of  hours  in  order  to  avoid  stopping,  and  thereby  caus- 
ing a  jog  in  the  cylinder. 


OF  THE  STEAM-ENGINE.  19 

advisable  to  make  a  steam-cylinder  of  less  than  0.75  inch 
thickness  under  any  circumstances. 

In  deciding  upon  the  thickness  to  be  given  to  any  cylin- 
der, the  method  of  fastening  it,  as  well  as  the  distorting 
forces  that  are  likely  to  occur,  should  be'  carefully  con- 
sidered. 

(5.)  Thickness  of  the  Cylinder -Heads.— It  is  demon- 
strated in  Weisbach's  Mechanics  of  Engineering,  Vol.  i.,  Sec. 
vi.,  Art.  363,  that  if  the  cylinder-heads  were  made  of  a 
hemispherical  shape,  they  would  need  to  be  of  only  half 
the  thickness  of  the  cylinder-walls ;  and  in  designing,  the 
attempt  is  sometimes  made  to  attain  greater  strength  by 
giving  to  the  cylinder-heads  the  form  of  a  segment  of  a 
sphere. 

In  considering  rectangular  plane  surfaces  subjected  to 
fluid  pressure  in  Weisbach's  Mechanics  of  Engin eering,  Vol. 
ii.,  Sec.  ii.,  Art.  412,  the  following  formula  is  deduced  for 
square  plane  surfaces,  which  will  of  course  be,  with  greater 
safety,  true  for  a  circular  inscribed  surface : 

Let  ti  =  the  thickness  in  inches  of  the  cylinder-heads. 
"  Pb  =  the  boiler-pressure  in  pounds  per  square  inch. 
"  d  =  the  diameter  of  the  steam-cylinder  in  inches. 

<!  =  0.003d  \Spt>  (7) 

In  large  cylinders  the  heads  are  stiffened  by  casting  radial 
ribs  upon  them. 

Example. — 

Let  d  =  20  inches. 
"   Pb  =  150  pounds  per  square  inch. 

We  have,  from  formula  (7), 

*!  =  0.003dl/^  =  0.003  x  201/150  =  0.73  inches. 
Comparing  this  with  the  result  derived  from  formula  (6),  we 


20  THE  RELATIVE  PROPORTIONS      • 

observe  it  to  be  less  ;  but  we  also  find,  comparing  formulae 
(7)  and  (6),  that 


.003^     P*       10 


Prom  formula  we  see  that  iv  =  t  when  Pb  =  100  pounds  ;  that 
$!  is  greater  than  t  when  Pb  <  100  pounds  ;  and  that  ^  is  less 
than  t  when  Pb  >  100  pounds. 

A  good  practical  rule  for  engines  in  which  the  pressure 
does  not  exceed  100  pounds  per  square  inch  is  to  make  the 
thickness  of  the  cylinder-heads  one  and  one-fourth  that  of 
the  steam-cylinder  walls. 

(6.)  Cylinder-Head  Bolts.  —  Having  assumed  a  conve- 
nient width  of  flange  upon  the  steam-cylinder,  the  diameter 

¥of  the  boll  should  be  assumed  at  one-half  that  width,  and 
thoroughfare  bolts  used  preferentially  to  stud  bolts,  as  a 
stud  bolt  is  likely  to  rust  and  stick  in  place,  and  be  broken 
off  in  the  attempt  to  remove  it. 

The  bolts  fastening  the  cylinder-head  to  the  cylinder 
should  not  be  placed  too  far  apart,  as  that  would  have  a 
tendency  to  cause  leaks. 

Taking  5000  pounds  per  square  inch  of  the  nominal  area 
of  a  bolt  as  the  safe  strain,*  in  order  to  cover  fully  the 
strain  upon  the  bolt  due  to  screwing  its  nut  firmly  home,  as 
well  as  the  strain  due  to  the  steam-pressure,  and  dividing 
the  total  pressure  of  the  steam  upon  the  cylinder-head  by  it, 
we  will  obtain  the  area  of  all  the  bolts  required  ;  and  divid- 

*  The  Baldwin  Locomotive  Works  use  eleven  |-inch  stud  bolts  to 
secure  the  head  of  an  18-inch  steam-cylinder.  If  we  assume  the 
greatest  steam-pressure  to  be  150  pounds  per  square  inch,  we  have 
for  the  stress  per  square  inch  of  nominal  area  of  the  bolts  about  5800 
pounds,  and  we  are  therefore  well  within  limits  which  have  been 
found  thoroughly  practical  by  a  tentative  process. 


OF  THE  STEAM-ENGINE.  21 

ing  this  latter  area  by  the  area  of  one  bolt  of  the  assumed 
diameter,  we  have  the  number  of  bolts  required. 

Let  P6  =  the  boiler-pressure  in  pounds  per  square  inch. 
"      d  =  the  diameter  of  the  steam-cylinder  in  inches. 
"      c  =  the  area  of  a  single  bolt  of  the  assumed  diameter 

in  square  inches. 

"      b  =the  number  of  bolts  required. 
We  have 


5000c 

Example.  —  In  a  given  steam-cylinder, 
Let   d  =  32  inches. 

"    Pb  =  81  pounds  per  square  inch. 

"      c  =  0.442  square  inches  (diameter  of  bolt  J  inch). 

We  have,  from  formula  (9), 

5  =  0.000^^30, 

showing  about  30  three-quarter  inch  bolts  to  be  required. 
With  so  wide  a  margin  as  is  given  by  the  assumption  of 
5000  pounds  per  square  inch  as  a  safe  strain,  considerable 
variation  may  be  made  from  this  number.  Mr.  Robert 
Briggs,  in  a  paper  in  the  Journal  of  the  Franklin  Institute 
for  February,  1865,  page  118,  says:  "Ordinary  wrought 
iron,  such  as  is  generally  used  in  bolts,  can  be  stated  to  be 
reliable  for  a  maximum  load  under  20,000  pounds  per 
square  inch,  and  the  absolute  (ultimate  ?)  tensile  strength  of 
any  bolt  may  be  safely  estimated  on  that  basis." 


22 


THE  RELATIVE  PROPORTIONS 


CHAPTER    II. 

(7.)  Standard  Screw-Threads  for  Bolts.— The  stand- 
ard American  pitch  and  dimensions  of  head  and  nut  of  bolts 
as  now  used  in  all  the  mechanical  workshops  of  the  United 
States  was  first  proposed  by  Mr.  Wm.  Sellers.  (See  Report 
of  Proceedings,  Journal  of  the  Franklin  Institute  for  May, 
1864 ;  Report  of  Committee,  Journal  of  the  Franklin  In- 
titute,  January,  1865.  For  a  full  critique  and  comparison 
with  other  systems,  see  Journal  of  the  Franklin  Institute, 
February,  1865;  On  a  Uniform  System  of  Screw-Threads, 
Robert  Briggs.) 

The  advantages  of  uniformity  of  dimensions  in  an  element 
of  a  machine  so  frequently  occurring  do  not  need  discussion. 

The  "  Committee  on  a  Uniform  System  of  Screw-Threads  " 
reported  as  follows : 

"Resolved,  That  screw-threads  shall  be  formed  with 
straight  sides  at  an  angle  to  each  other  of  60°,  having  a 
flat  surface  at  the  top  and  bottom  equal  to  one-eighth  of  the 
pitch.  The  pitches  shall  be  as  follows,  viz. : 


Diameter  of  bolt.. 
No.  threads  pr.  in. 


Diameter  of  bolt.. 
No.  threads  pr.  in. 


A 


18 


16 


14 


13 


3* 


12 


31- 


11 


3' 


10 


4* 


21 


H 


4f 


1 


21 


Ifll! 


5| 


2| 


"  The  distance  between  the  parallel  sides  of  a  bolt  head 
and  nut,  for  a  rough  bolt,  shall  be  equal  to  one  and  a  half 
diameters  of  the  bolt  plus  one-eighth  of  an  inch.  The  thick- 
ness of  the  heads  for  a  rough  bolt  shall  be  equal  to  one-half 
the  distance  between  its  parallel  sides.  The  thickness  of  the 
nut  shall  be  equal  to  the  diameter  of  the  bolt.  The  thick- 
ness of  the  head  for  a  finished  bolt  shall  be  equal  to  the 


OF  THE  STEAM-ENGINE. 


23 


thickness  of  the  nut.  The  distance  between  the  parallel 
sides  of  a  bolt  head  and  nut  and  the  thickness  of  the  nut 
shall  be  one-sixteenth  of  an  inch  less  for  finished  work  than 
for  rough." 


FIG.  2. 


The  required  dimensions  of  bolts  and  nuts  can  be  best 
expressed  in  a  general  way  by  means  of  formulae  and 
Fig.  2. 


24  THE  RELATIVE  PROPORTIONS 

Let   D  —  the  nominal  diameter  of  any  bolt  in  inches. 
Then  p  =  the  pitch  =  0.24V/Z)  + 0.625  -  0.175. 

n  =  the  number  of  threads  per  inch  =  -. 

P 
d  =  the  effective  diameter  of  the  bolt — i.  e.,  at  root 

of  thread  =  D-1.3p. 
£=the  depth  of  thread  =  0.65p. 
H=  the  depth  of  nut  =  D. 
dn  =  the  short  diameter  of  hexagonal  or  square  nut 

=  -Z>+0".125. 

A  =  the  depth  of  the  head  of  bolt  =  |D+TV  inch. 

g 
dh  =  the  short  diameter  of  the  head  of  bolt  =  -D  +  0.125. 

The  value  of  the  pitch  p,  in  terms  of  D,  was  derived  from 
a  graphical  comparison  of  the  then  existing  threads  as  used 
in  the  most  prominent  workshops  in  the  United  States.  The 
depth  of  the  thread  S  is  deduced  as  follows : 

Since  the  angles  of  a  complete  V  thread  are  each  =  60°, 
its  sides  and  the  pitch  would  form  an  equilateral  triangle, 
and  we  would  have  for  its  depth  p  sin  60°  =  0.866p  ;  but  in 
the  actual  thread  £  is  taken  off  at  the  top  and  bottom;  leav- 
ing only  J  of  the  depth  of  a  complete  V  thread. 

Therefore,  S  -  -QMGp  -  0.65p. 

4 

Were  it  possible  or  always  convenient  to  have  uniformly 
close  work,  such  an  accident  as  the  stripping  of  the  thread 
from  a  bolt  with  the  dimensions  stated  would  be  impossible. 

The  proportions  established  are  the  result  and  an  average 
of  the  practical  requirements  of  machine-shop  practice,  and 
are  therefore  to  be  preferred  to  proportions  which  might  be 
established  from  a  theoretical  consideration  of  the  strength 


OF  THE  STEAM-ENGINE.  25 

of  materials.     The  appended  table  of  dimensions  (pp.  26, 
27)  is  furnished  by  Messrs.  "Wm.  Sellers  &  Co. 

(8.)  The  Steam-Chest. — In  deciding  upon  the  dimen- 
sions of  the  steam-chest,  it  must  be  borne  in  mind  that  it 
ought  to  be  as  small  as  the  dimensions  and  travel  of  the 
valve  will  permit,  in  order  to  avoid  loss  by  condensation 
of  steam. 

The  chest  is  subject  to  many  modifications  of  form,  but 
usually  consists  of  the  ends  and  two  sides  of  a  cast-iron  box, 
resting  upon  a  rim  surrounding  the  valve-face,  upon  which 
a  flat  cover  is  placed,  and  the  whole  firmly  secured  to  the 
steam-cylinder  by  means  of  stud-bolts  passing  through  the 
cover  outside  of  the  sides  of  the  box,  in  order  to  avoid 
rusting  of  the  bolts  as  well  as  to  diminish  the  contents  of 
the  box. 

The  number  of  bolts  required  can  be  determined,  as  shown 
in  Art.  (6),  from  a  consideration  of  the  steam-pressure  upon 
the  steam-chest  cover. 

It  is  customary  to  make  the  sides  and  cover  of  the  steam- 
chest  of  the  same  material  and  thickness  as  the  cylinder- 
walls,  sometimes  strengthening  the  cover  by  casting  ribs 
upon  it. 

To  deduce  the  theoretical  thickness,  we  have,  from  Weis- 
bach's  Mechanics  of  Engineering,  Vol.  ii.,  Sec.  ii.,  Art.  412, 
the  following  formula : 

Let  I   =  the  longest  inside  measurement  of  chest  in  inches. 
"     b  =  the  breadth  of  chest  in  inches. 
"    Pb  -  the  boiler-pressure  in  pounds  per  square  inch. 
"    T  -  the  safe  tensile  strain  upon  cast  iron  per  square 

inch  - 1800  pounds. 
"    <2  =  the  required  thickness  of  the  steam-chest  cover  in 

inches. 


THE  RELATIVE  PROPORTIONS 


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PL. 
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55 


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OF  THE  STEAM-ENGINE. 


27 


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28  THE  RELATIVE  PROPORTIONS 

« 


Example.  —  In  a  given  steam-chest, 

Let  f  =  40". 
"  5  =  20". 
"  Pb  =  81  pounds. 

Then,  substituting  in  formula  (10),  we  have 


.     40    /       1600x400       C1     Q  .    , 

/2  =  — -v  / 81  =  3  inches. 

60  \  2560000  +  160000 

(9.)  The  Steam-Ports.*— D.  K.  Clark,  in  Railway 
Machinery,  page  108,  states  that  with  a  piston-speed  of  600 
feet  per  minute  a  port  area  equal  to  one-tenth  of  the  piston 
area  is  found  sufficient  to  admit  steam  to  the  cylinder  with 
sufficient  facility  for  all  practical  purposes,  and  with  but  a 
slight  reduction  of  pressure.  The  size  of  the  port  may  be 
increased  or  diminished  proportionally  to  the  piston-speed 
from  these  data  with  good  success,  bearing  in  mind  that  all 
tortuousness  of  direction  that  can  be  should  be  avoided  in 
ports. 

Example. — Let  the  piston-speed  =  500  feet  per  minute ; 

*  For  a  thorough  treatise  on  link-  and  valve-motions,  as  well  as  in- 
dependent cut-offs,  see  Zeuner's  Treatise  on  Valve-Gears.  Auchin- 
closs  (Link-  and  Valve-Motions)  treats  the  same  subject  in  a  simpler 
manner,  avoiding  the  use  of  formulae  as  much  as  possible.  King 
(Steam,  Steam-Engine  Propellers,  etc.)  gives  good  descriptions  of  some 
purely  American  forms  of  valves.  Burgh  (Link- Motions)  gives  a 
large  number  of  examples  of  English  link-motions  which  have  been 
constructed.  The  author  refrains  from  entering  upon  the  wide  field 
of  valve-motions,  as  one  which  has  been  thoroughly  covered  by  the 
splendid  work  of  Zeuner  mentioned  above. 


OF  THE  STEAM-ENGINE.  29 

then  the  port  area  should  be  =  f  x  ^  piston  area  =  TV  piston 
area. 

(10.)  The  Piston-Head.— It  is  very  doubtful  if  any 
formula  can  be  derived  which  will  give  a  correct  value  for 
the  thickness  of  the  piston-head  under  all  circumstances. 
If  the  steam-cylinder  be  laid  horizontally,  other  things  being 
equal,  the  piston-head  should  be  broader  than  if  the  cylinder 
is  vertical,  and  an  extra  breadth  of  piston  should  be  given 
in  all  cases  of  rough  usage  or  very  rapid  piston-speed. 

The  writer  offers  as  a  guide  only  the  following  formula : 

Let  -L  =  the  length  of  stroke  in  inches. 
"     d  =  the  diameter  of  cylinder  in  inches. 
"     £3  =  the  thickness  of  the  piston-head  in  inches. 


Then  (.-v'Zar,  (11) 

or  if  L  =  d,  (,-j/J.  (12) 


Example. — Let  L  =  24  inches. 
"    d  =  20  inches. 

Then,  ^  =  y  24  x  20  =  4J-  inches,  approx.  ]/ 

(11.)  The  Piston-Rod. —  The  piston-rods  of  the  best 
engines  are  made  of  steel,  although  in  many  instances  ham- 
mered wrought  iron  is  still  used.  The  rod,  being  fastened  at 
one  end  to  the  piston-head  and  at  the  other  keyed  to  the 
cross-head,  sustains  an  alternate  thrust  and  pull  while  pass- 
ing accurately  through  the  stuffing-box  and  gland  at  one 
end  of  the  cylinder. 

It  is  obvious  that  the  rod  must,  in  addition  to  being  strong 
enough  to  sustain  these  stresses,  be  possessed  of  sufficient 
rigidity  not  to  bend  in  the  slightest  degree  while  in  action. 
3  * 


30  THE  RELATIVE  PROPORTIONS 

As  the  piston-head  is  aided  in  holding  its  position  acros3 
the  cylinder  by  means  of  the  rod,  and  the  cross-head  is 
always  liable  to  have  a  slight  amount  of  lateral  play,  due 
to  imperfections  of  workmanship  or  wearing  of  the  guides, 
we  must  regard  it  as  a  solid  column  fixed  at  one  end  and 
loaded  at  the  other,  which  is  free  to  move  sideways. 

(12.)  Wrought-Iron  Piston-Rod, — For  the  purpose  of 
determining  the  proper  diameter  of  the  piston-rod,  either  of 
the  two  following  formulae  may  be  used,  both  taken  from 
Sec.  iv.  of  Weisbach's  Mechanics  of  Engineering : 

Let    $=the  stress  upon  the  whole  piston-head  due  to  the 

pressure  of  the  steam. 
"     di  =  the  diameter  of  the  rod  in  inches. 
"      .F=the  cross-section  of  the  piston-rod  in  square  inches. 
"     -BT=the  ultimate  crushing  strength  per  square  inch  of 

wrought  iron  =  31000  pounds  (Weisbach). 
"     W=  the  measure  of  the  moment  of  flexure  of  a  round 


"     J£  =  the  modulus   of    elasticity   of    wrought    iron  = 

28000000  pounds  per  square  inch. 
"       I  =  the  length  of  the  rod  in  inches  =  L  =  the  stroke. 

We  have,  for  columns  breaking  by  direct  crushing, 

S=FK,  (13) 

and  for  long  columns  tending  to  break  by  bending  —i.  e., 
buckling — 

(14) 

Formula  (13)  must  be  used  when  the  resulting  diameter 
of  the  rod  is  greater  than  one-twelfth  of  its  length.     If  the 


OF  THE  STEAM-ENGINE.  31 

use  of  formula  (13)  results  in  a  diameter  less  than  one-twelfth 
the  length  of  the  rod,  formula  (14)  must  be  used.* 

If  we  assume  5  as  the  least  allowable  factor  of  safety  for 
materials  subjected  to  a  constant  strain  in  one  direction  only, 
then  will  10  be  the  proper  factor  for  members  subject  to 
alternate  and  equal  stresses  in  opposite  directions  —  i.  e.t  ten- 
sion and  compression.  (See  Weyrauch,  Structures  of  Iron 
and  Steel,  translated  by  Du  Bois,  chapters  iv.  and  xiii.f) 

As  before, 

Let  Pb  =  the  steam-pressure  in  boiler  in  Ibs.  per  square  inch. 
"      d  =  the  diameter  of  the  steam-cylinder  in  inches. 

The  greatest  stress  which  the  piston-rod  has  to  sustain  is 
that  due  to  the  initial  pressure  of  the  steam  in  the  cylinder, 
which  can  be  taken  as  equal  to  the  boiler-pressure. 

Introducing  a  factor  of  safety  of  10  and  substituting  in 
formula  (13),  we  have, 

*  To  determine  the  ratio  of  the  length  to  the  diameter  of  a  wrought- 
iron  rod  at  which  its  tendency  to  crush  is  equal  to  its  tendency  to 
break  by  buckling,  we  place  formula  (13)  =  formula  (14).  Let  d  = 


ftd'*         I  TT  \  2  Trd* 
the  diameter  of  rod  in  inches,     ^-K=(  —  j  -  E.      Therefore, 

—  =  —  A/—  ,  and  substituting  the  values  of  .E'and  K,  we  have  —  =  12. 

f  The  ever-varying  chemical  constituents  of  the  various  brands 
of  wrought  iron  ;  the  fact  now  fully  ascertained  that,  however  care- 
fully made  and  worked,  wrought  iron  may  be  heterogeneous  in  its 
composition  ;  that  the  reduction  of  the  area  from  pile  to  bar  has  a 
very  great  influence  upon  its  strength  (the  greater  the  reduction, 
the  greater  its  strength)  ;  that  a  high  tensile  and  compressive 
strength  is  often  obtained  by  loss,  to  a«  great  extent,  of  ductility 
and  welding  power  —  all  being  considerations  of  a  purely  practical 
nature,  as  well  as  the  theoretical  one  mentioned  above,  —  point  to  a 
higher  factor  of  safety  than  6  or  8,  which  is  generally  recommended. 


32  THE  RELATIVE  PROPORTIONS 

lOP^L  ^31000. 
4        4 

Therefore,  4  =  0.0179(^2* 

Or  supposing  the  pressure  Pb  uniform  throughout  the  stroke, 
we  have  from  formula  (1) 

*f    33000(gP) 
~          ~ 


if  we  take  the  stroke  in  inches.     Therefore,  from  formula  (15) 
(HP)  33000x12x4 

~~~ 


Therefore,  d,  =  12.753  \  (16) 

*  LN 

Note  that  the  stroke  is  taken  in  inches. 
.  Substituting  in  formula  (14)  the  values  as  stated  above, 
we  have,  using  a  factor  of  safety  of  10, 


64 


Therefore,  4  =  J — r/WI7Pb  = 

\  TT'X  28000000  v 

0.03901  {/WI7Fb  inches.  (17) 

If  we  take  L  =  d,  formula  (17)  becomes 

4  =  0.03901d  i/Pb  inches.  (18) 

If  we  suppose  the  pressure  of  the  steam  to  be  uniform, 
formula  (17)  becomes,  in  terms  of  the  horse-power  (JJP), 


!  - 1.039^  (gP)     inches.  (19) 


OF  THE  STEAM-ENGINE.  33 

From  the  consideration  of  the  preceding  formulae  we 
deduce  the  following  rule  for  wrought-iron  piston-rods: 

Deduce  the  diameter  of  the  piston-rod  by  means  of  formula 
(15)  or  (16),  as  may  be  most  convenient.  Should  this  diameter 
be  less  than  one-twelfth  of  the  stroke,  then  use  the  most  conve- 
nient of  formulas  (17),  (18),  and  (19). 

Example. — What  should  be  the  diameter  of  a  piston-rod 
for  a  steam-cylinder  ?  Data  as  follows : 

L  =  48  inches. 
d   =  32  inches. 

P  =  40  pounds  per  square  inch. 
N  =  40  per  minute. 
(.HP)  =  156,  approx. 

Using  formula  (15),  we  have 

*  -  0.0179  x  32 i/30  -  3.6  inches. 
Using  formula  (16),  we  have 

dl  - 12.753  J  156    =  3.63  inches. 
*  48  x  40 

As  the  diameter,  3.6  inches,  is  less  than  one-twelfth  of  the 
stroke,  48  inches,  we  must  use  formula  (17)  or  (19). 
Substituting  in  formula  (17),  we  have 

dl  =  0.03901 1//40x322x482  =  3.84  inches. 
Substituting  in  formula  (19),  we  have 

d,  =  1.039  ^— —  =  3.84  inches. 

NOTE. — It  must  be  borne  in  mind  that  the  piston-rod  will 
almost  always  be  proportioned  from  the  initial  or  boiler- 


34  THE  RELATIVE  PROPORTIONS 

pressure  of  the  steam,  Pb,  and  not  from  the  mean  pressure, 
P,  as  in  all  the  better  forms  of  engine  the  mean  pressure  is 
much  less  than  the  initial  pressure. 

(13.)  Steel  Piston-Rod. — When  steel  is  the  material 
used  for  piston-rods,  its  greater  modulus  of  ultimate  strength 
and  differing  coefficient  of  elasticity  will  alter  the  numerical 
values  in  the  formulae  given  above. 

Hodgkinson,  in  his  experiments  on  long  columns  of 
wrought  iron  and  steel,  found  that  a  steel  column  of  the 
same  dimensions  as  a  wrought-iron  column  would  bear  one 
and  one-half  times  as  great  a  load. 

A  consideration  of  formula  (14)  shows  that  these  experi- 
ments would  require  that  E,  the  modulus  of  elasticity  for 
steel  (which  is  28,000,000  pounds  for  wrought  iron),  should 
become  42,000,000  pounds. 

This  value  agrees  very  nearly  with  that  given  by  Reu- 
leaux  in  Der  Constructeur,  page  4,  for  cast  steel. 

Kupffer,  Styffe  and  Fairbairn  place  the  value  of  E  at 
between  30  and  31  million  pounds  per  square  inch  for  steel. 

As,  however,  we  shall  use  the  value  E=  42,000,000  Ibs. 
under  exactly  the  same  conditions  as  the  experiments  from 
which  it  is  derived,  it  is  probably  the  most  correct  value  for 
our  purposes. 

The  crushing  strength  of  steel  is  so  much  greater  than  its 
tensile  strength,  where  the  material  is  not  permitted  to  de- 
flect, that  it  need  not  be  taken  into  consideration  in  formula 
(13). 

As  a  mean  of  eleven  experiments  made  by  Kirkaldy  on 
steel  taken  at  random  from  merchants'  stores,  in  all  cases 
the  extension  being  upward  of  10  per  cent.,  we  have  for  the 
tensile  breaking  strain  (equal  to  K)  about  90,000  pounds 
per  square  inch. 

The  elongation  of  10  per  cent,  or  upward  would  indi- 
cate steel  of  sufficient  toughness  for  machinery  purposes. 


OF  THE  STEAM-ENGINE.  35 

In  the  absence  of  reliable  experiments  determining  the 
ultimate  crushing  strength  of  soft  steel,  we  are  safe  in  as- 
suming that  it  will  bear  100,000  pounds  per  square  inch 
without  deformation,  which  assumption  makes  the  crushing 
strength  of  steel  equal  to  that  of  cast  iron,  and  which  it 
probably  much  exceeds. 

Substituting  in  formula  (13)  and  using  the  same  notation 
as  in  Art.  (12),  we  have 


4  4 

Therefore,     ^  =  (M)105dv<F6.  (20) 

Or  if  we  suppose  the  pressure  uniform  throughout  the 
whole  stroke,  we  have,  in  terms  of  the  horse-power, 


(21) 


LN 

Substituting  in  formula  (14),  we  have 


=42000000,. 


4          4L*         64 

and  reducing,  we  have 

d1-0.03525|^3IZ^;  (22) 

or  if  in  (22)  we  suppose  L  =  d, 

(23) 


If  we  suppose  the  initial  pressure  to  be  uniform  through- 
out the  whole  stroke,  we  have 

(24)  ' 


36  THE  RELATIVE  PROPORTIONS 

Formulae  (20)  and  (21)  should  be  used  when  the  result- 
ing diameter  of  the  piston-rod  is  greater  than  J-  of  the  length 
of  the  stroke  ;  if  it  is  less,  then  formula  (22),  (23)  or  (24) 
must  be  used.* 

Example.  —  What  should  be  the  diameter  of  a  steel  piston- 
rod  for  steam-cylinder  ?  Data  as  before. 

L  =  48  inches. 
d  =  32  inches. 

P  =  40  pounds  per  square  inch. 
N=  40  per  minute. 
=  156approx. 


Using  formula  (20),  we  have  ^  =  0.0105x32/40  =  2.12 

I      -t  en 

inches.    Using  formula  (21),  we  have  di  =  7.48^/  — 

»  4o  x  4U 

2.13  inches. 

We  observe  that  the  diameter  of  rod  resulting  from  form- 
ulae (20)  and  (21)  is  less  than  -|-  of  the  stroke,  and  we  must 
therefore  use  formula  (22)  or  (24). 

Substituting  in  formula  (22),  we  have 


<?!  =  0.03525|/322  x  482  x  40  =  3.47  inches. 
Substituting  in  formula  (24),  we  have 

d,  =  0.9394^/156x48  -  3.47  inches. 
*  Referring  to  note  to  Art.  (12),  we  find  the  following  general 

/    TT  n? 

formula:  ——  ^\'-=.    Substituting  in  this  the  assumed  values  of  E 
a     o  >  K 

i    .  E=  42000000        ,         I 
and  Kfor  steel,  ,  we  have  -  =  8. 


OF  THE  STEAM-ENGINE.  37 


CHAPTER    III. 

14.  Comparison  and  Discussion  of  Wrought-Iron 
and  Steel  Piston-Rods. — If,  considering  the  formulse  for 
rupture  by  crushing  or  tearing  only  for  wrought  iron  and 
steel,  we  divide  (21)  by  (16),  we  have 


For  steel  c^  7.- 

= 1_^L  =  0.58.    (25) 

I7HF) 
For  wrought  iron  dl     12.75^ 

That  is,  the  steel  rod  would  have  but  0.58  the  diameter 
of  the  wrought-iron  rod,  and  0.34  the  area. 

If  we  treat  the  formulse  (24)  and  (19)  for  rupture  by 
buckling  in  a  similar  manner,  we  have 


For  steel  di 

\       i\ 

0.90.    (26) 

For  wrought  iron  d+ 


That  is,  the  steel  rod  has  0.9  the  diameter  and  0.81 
the  area  of  a  wrought-iron  rod  working  under  the  same 
conditions. 

In  either  case  the  use  of  steel  is  productive  of  economy 
in  weight,  varying  from  34  to  81  per  cent. 

Letting  V=  the  volume  of  the  piston-rod  in  cubic  inches, 
"       y  =  the  weight  of  a  cubic  inch  of  wrought  iron 
or  steel  =  0.27  lb.,  approx., 

we  have,  for  purposes  of  comparison, 


38  THE  RELATIVE  PROPORTIONS 

F-^Z.  (27)* 

Letting  C  represent  the  numerical  constant,  we  have,  by 
substituting  for  d?  its  value,  derived  equation  (16)  or  (21) 
for  crushing, 


and  we  see  from  (28)  that  the  volume,  and  consequently 
the  weight,  of  a  piston-rod  are  inversely  as  the  number  of 
strokes  per  minute  for  any  assigned  horse-power. 

Substituting  formulae  (19)  and  (24)  for  rupture  by  buck- 
ling in  (27),  we  have 


If  in  this  we  assume  L  =  d,vre  have  from  equation  (2), 
letting  Ci  represent  the  constant  in  that  equation, 


giving  an  expression  similar  to  (28),  but  affected  by  the 
steam-pressure,  and  showing  an  economy  in  weight  to  be 
derivable  from  high  pressure  as  well  as  piston-speed  for  this 
particular  case  —  that  is,  when  the  stroke  and  diameter  of 
the  steam-cylinder  are  equal,  and  the  diameter  of  the  rod 
does  not  exceed  for  wrought  iron  ^  and  for  steel  £  of  its 
length. 

If  we  multiply  the  results  of  formula  (28),  (29)  or  (30) 
by  y  =  0.27,  and  by  the  factor  representing  the  ratio  of  the 

*  This  statement  is  not  accurately  true,  as  the  piston-rod  must  be 
somewhat  longer  than  the  stroke.  Multiplying  (27)  by  a  factor  of 
from  |  to  2  would  give  an  approximate  result. 


OF  THE  STEAM-ENGINE.  39 

total  length  of  the  piston-rod  to  the  stroke,  we  obtain  its 
weight  in  pounds  for  a  uniform  pressure. 

Referring  to  formula  (15)  or  (20)  for  crushing  or  tearing, 
and  to  formula  (17)  or  (22)  for  rupture  by  buckling,  we  see 
that  the  diameter  of  the  piston-rod  increases  with  the  square 
root  of  the  pressure,  or  its  area  increases  directly  as  the 
pressure  of  the  steam  in  the  first  case — i.  e.,  for  crushing  or 
tearing — and  that  the  diameter  of  the  piston-rod  increases 
with  the  fourth  root  of  the  steam-pressure,  or  its  area  with 
the  square  root  of  the  pressure,  in  the  second  case. 

All  of  these  formulae  alike  show  that  a  rapid  increase  in 
the  boiler-pressure  does  not  cause  a  correspondingly  rapid 
increase  in  the  diameter  of  the  rod,  and  explain  the  success 
of  empirical  rules  giving  a  constant  ratio  between  the  diam- 
eters of  piston-rods  and  steam-cylinders  regardless  of  the 
steam-pressure. 

For  the  purpose  of  showing  how  little  variation  in  the 
diameters  of  piston-rods  results  from  great  changes  in  the 
steam-pressure,  the  'following  short  table  of  second  and 
fourth  roots  of  the  usual  steam-pressures  is  given : 

TABLE  II. 


Number. 

Square  root. 

Fourth  root. 

25 
50 
75 
100 
125 

5.00 
7.07 
8.66 
10.00 
11.18 

2.236 
2.659 
2.949 
3.163 
3.344 

Thus  we  see  that  for  a  pressure  increased  5  times  formulae 
(15)  and  (20)  increase  the  diameter  of  the  rod  but  a  little 
over  2  times,  and  formulae  (17)  and  (22)  increase  the  diam- 
eter of  the  rod  but  a  little  less  than  one-half. 

Generally  it  will  be  found  that  the  formula  for  crushing 
(15)  will  give  the  greatest  diameter  for  a  wrought-iron 


40  THE  RELATIVE  PROPORTIONS 

piston-rod,  and  that  formula  (22)  will  give   the  greatest 
diameter  of  a  steel  piston-rod. 

(15.)  Keys  and  Gibs.  —  Reserving  keys  for  shafts  for 
discussion  in  Article  (40),  we  will  consider  that  form  of 
key  used  in  making  the  connection  between  the  piston-rod, 
piston-head  and  cross-head,  and  also  for  the  connecting-rod. 

If  we  wish  to  avoid  weakening  the  piston-rod  at  the  point 
where  the  key  passes  through  it  —  a  precaution  not  found  to 
be  practically  necessary  in  most  cases  —  we  must  increase  the 
diameter  of  the  rod  at  that  point. 

It  is  customary  to  thicken  the  rod  where  it  enters  the 
piston-head,  but  for  convenience  not  to  do  so  where  the 
piston-rod  enters  the  cross-head.  If  it  is  desired  to  thicken 
the  rod  at  both  ends,  it  is  best  to  make  it  of  a  uniformly 
enlarged  diameter  from  end  to  end. 

!Let  d'  =  the  diameter  of  the  enlarged  end  of  rod  in  inches. 
"    di  =  the  diameter  of  the  piston-rod,  as  derived  in  Art. 
(12),  (13)  or  (14),  in  inches. 

The  average  dimensions  of  keys  are  assumed  as  follows  : 

h  =  the  breadth  of  the  key    =  df. 

t  =  the  thickness  of  the  key  =  —  . 

4 

d'9 
Therefore,    2ht  =  —  ,  which  nearly  equals  the  cross-section 

2i 

of  the  enlarged  end  -        -      =  0.5354<r. 


Bearing  in  mind  that  there  are  two  shearing-sections  for 
every  through-key,  and  assuming  the  shearing  strength  of 
wrought  iron  equal  to  its  tensile  or  compressive  strength,  as 
is  customary  in  practice. 


OF  THE  STEAM-ENGINE. 


41 


We  can  place,  if  the  enlarged  end  of  the  rod  weakened 
by  the  key-way  and  the  key  are  to  be  of  equal  strength, 


444 

Reducing,  2.141 6<T  =  3.1416^. 
Therefore,  <T=  1.2114. 


(31) 


These  various  dimensions  and  measurements  are  shown  in 
Fig.  3. 

FIG.  3. 


If  one  or  two  gibs  are  used  in  connection  with  the  wedge- 
form  of  key,  their  mean  area  must  be  used  in  the  calcula- 
tions as  for  a  single  key.  Various  methods  are  used  to 
prevent  keys  from  falling  out,  as  split  pins,  set  screws,  and 
bolts  attached  to  one  end  of  the  key  and  held  by  a  nut  on 
the  gib. 

(16.)  Wrought-Iron  Keys. — The  method  given  in  the 
preceding  paragraph,  although  the  usual  one  in  practice,  is 
liable  to  cause  some  error  in  results. 

4* 


42  THE  RELATIVE  PROPORTIONS 

Let  jP=the  mean  area  of  a  key  in  square  inches. 
"     K=  the  shearing  strength  (ultimate)  of  wrought  iron 

per  square  inch  =  50000  pounds. 
"    Pb  =  the  boiler-pressure  in  pounds  per  square  inch. 
"      d  =  the  diameter  of  the  steam-cylinder  in  inches. 
"    10  -  the  factor  of  safety. 

Kecollecting  that  every  through-key  has  two  shearing  sur- 
faces, we  have 


4 
Therefore,  F=  0.000078e*2P6.  (32) 

Or,  if  we  assume  the  steam-pressure  to  be  uniform  through- 
out the  stroke,  and  take  the  length  of  stroke  in  inches,  we 
have 

/•  i"T"rv\. 

(33) 


, 

and  placing  2F=  —  ---  —  , 

we  have         ^-1.9  (34) 


Example.  —  Let       P6  =the  boiler-pressure  =  the  pressure 

throughout    the    stroke  =  40 
pounds  per  square  inch. 
"  d  =  32  inches. 

"          L=  48  inches. 
«         jy  =  40  per  minute. 
"    (J3P)  =  156,  approx. 
From  formula  (32)  we  have 

F=  0.000078  x  1024  x  40  =  3.19  square  inches. 
From  formula  (33)  we  have 


OF  THE  STEAM-ENGINE.  43 

"1  E\£ 

.F=39.6  -  =  3.21  square  inches. 
48x40 

Therefore,  from  formula  (34,),  we  have 
d'  -  1.9  /33  =  3.40  inches. 


Comparing  d'  =  3.40  inches  with  dl  in  the  example  at  the 
end  of  Art.  12,  we  see  that  we  are  able  to  key  the  rod  with- 
out thickening  it  at  either  end,  if  we  suppose  the  safe  shear- 
ing and  tensile  strength  equal,  and  —  5000  pounds  per 
square  inch. 

(17.)  Steel  Keys.  —  Adopting  the  same  notation  as  in 
the  preceding  paragraph,  we  have  only  to  change  the  value 
of  K.  The  shearing  strength  of  steel  is  equal  to  three- 
fourths  of  its  tensile  strength  ;  therefore,  K=  f  90000  = 
67500  pounds  per  square  inch  area. 

We  have        2  x  6750  x  JP=  -dlP6. 

4 

Therefore",  F=  0.000058d2P6.  (35) 

Or  if  we  suppose  the  steam-pressure  to  be  uniform  through- 
out the  length  of  the  stroke,  and  take  the  stroke  in  inches, 


(36) 


Comparing  formulae  (33)  and  (36),  we  have 

For  steel,  F  __  29.33          . 
For  wrought  iron,  F~  39.6  " 

and  therefore  that  the  mean  area  of  a  steel  key  is  74  per 
cent,  of  the  mean  area  of  a  wrought-iron  key. 

Bearing  in  mind  that  the  shearing  strength  of  steel  is  but 
three-fourths  of  its  tensile  strength,  we  can  place  Art.  (15), 


44  THE  RELATIVE  PROPORTIONS 


Therefore,  d'  =  1.68  yT.  (37) 

Example.  —  Let       Pt  =  40  pounds  per  square  inch  =  uni- 

form pressure. 
"          d  =32  inches. 
"        L  =48  inches. 
"        N  =  40  per  minute. 
"  (J7P)  =  156,  approx. 

We  have,  from  formula  (35), 

F=  0.000058  x  1024  x  40  =  2.37  square  inches. 
We  have  also,  from  formula  (36), 

F=  29.33^^  --  2.37  square  inches. 
48x40 

From  formula  (37)  we  have 

d'  =  1.68  /237  -  1.73  inches. 


Comparing  this  latter  result  with  the  value  of  dl  in  the 
example  Art.  13  =  3.47,  we  find  that  no  enlargement  of  the 
ends  of  the  rod  is  needed. 

(18.)  The  Cross-Head.—  The  cross-head—  or  motion- 
block,  as  it  is  sometimes  called  —  to  one  end  of  which  the 
piston-rod  is  keyed,  and  extending  laterally  from  which  are 
the  slides  or  slide,  which  by  pressure  upon  the  guides  pre- 
serve the  rectilinear  motion  of  the  end  of  the  piston-rod  after 
leaving  the  cylinder,  and  to  the  other  end  of  which  the  con- 
necting-rod is  attached  by  means  of  a  pin,  usually  of  the 
same  diameter  as  the  crank-pin,  though  not  necessarily  so, 
has  many  different  forms. 

The  proportions  of  the  cross-head  or  motion-block  are 
with  a  few  exceptions  rather  a  matter  of  experience  and 
good  taste  than  of  calculation.  Reuleaux,  in  Der  Con- 


OF  THE  STEAM-ENGINE. 


45 


strueteur,  page  546  et  seq.,  gives  many  very  good  examples 
of  forms  of  cross-head.     A  few  of  the  more  common  forms 


are  shown  in  Fig.  4,  a,  b  and  c,  although  the  determination 


46  THE  RELATIVE  PROPORTIONS 

of  dimensions,  rather  than  the  suggestion  of  ingenious 
arrangements,  is  the  purpose  of  this  work.  Arthur  Rigg, 
in  A  Practical  Treatise  on  the  Steam-Engine,  suggests  (Plate 
41)  one  or  two  good  forms  of  cross-head. 

The  material  used  in  the  construction  of  cross-heads  is, 
according  to  circumstances,  cast  or  wrought  iron  or  steel. 

Example  a  is  a  form  of  cross-head  used  for  direct-acting 
pumping  and  blowing  engines,  and  its  dimensions  can  be 
calculated  from  the  formula  for  a  beam  fixed  at  one  end 
and  loaded  at  the  other,  given  in  Table  VI.,  the  load  being 
one-half  the  stress  due  to  the  steam-pressure  upon  the 
piston-head. 

Example  b  is  a  common  form  for  engines  having  but  two 
guides. 

Example  c,  sometimes  called  the  slipper  form,  having  but 
one  guide,  is  well  adapted  to"  engines  running  in  one  direc- 
tion only. 

Many  other  forms  of  cross-head  exist,  and  in  fact  almost 
every  new  construction  demands  special  adaptation  of  the 
form  of  cross-head. 

(19.)  Area  Of  the  Slides. — When  a  horizontal  engine 
"  throws  over  "  while  the  piston-head  is  moving  toward  the 
main  shaft,  all  of  the  pressure  and  wear  comes  upon  the  lower 
slide.  If  the  motion  of  the  engine  be  reversed,  all  of  the 
strain  and  wear  comes  upon  the  upper  slide. 

Since  lubricants  spread  and  flow  over  the  lower  guide 
more  easily  than  the  upper,  and  since  it  is  easier  to  resist  a 
strain  in  compression  than  tension  by  means  of  fastenings  to 
the  bed-plate,  it  is  customary  and  proper  to  cause  engines 
having  motion  in  one  direction  only  to  throw  over  rather 
than  under.  The  slipper-guide  (see  Fig.  4,  c)  can  be  used 
in  this  case  with  great  propriety. 

The  greatest  pressure  upon  a  slide  occurs  when  it  is  near 
the  middle  of  its  stroke  (assumed  at  the  middle  for  conve- 


OF  THE  STEAM-ENGINE.  47 

nience),  and  can  be  deduced  with  little  trouble  from  the 
triangle  of  forces. 

Let        S  -  the  total  steam-pressure  on  the  piston-head  in 
pounds  =  -d?Pb. 

"         Si  =  the  pressure  upon  the  guide  in  pounds. 

/  =  the  length  of  the  connecting-rod  in  inches. 

r  ?=  the  length  of  the  crank  =  —  in  inches. 

ft 

n  =  -  =  the  rates  of  the  length  of  the  connecting- 
rod  to  the  crank. 

Pb  =  the  boiler-pressure  in  pounds  per  square  inch. 
d  =  the  diameter  of  the  steam-cylinder  in  inches. 
L  =  the  length  of  stroke  in  inches, 
"    (HP)  =  the  indicated  horse  power. 

Referring  to  Fig.  5, 

FIG.  5. 


we  have     S  :  $  : :  T/  nV  -  r2  :  r. 

Therefore,     fl  -       S      .  (38) 

Vn*-\ 

The  value  of  n  commonly  varies  between  4  and  8. 
For  n  =  4,  formula  (38)  becomes 

"  0.2582& 
1/16-1 


48  THE  RELATIVE  PROPORTIONS 

For  n  =  8,  it  becomes 

4--=L=#-o.i2aa 

1/64-1 

If  in  this  we  substitute  the  value  S=-d2Pb,  we  have 

4 

3,0.7854gR  (39) 

I/  n2-! 

Or,  supposing  the  pressure  per  square  inch,  P6,  to  be  uni 
form  throughout  the  stroke,  we  have 

3%000_(flP) 


"We  have,  then,  from  either  equation  (39)  or  (40)  the 
maximum  pressure  upon  the  slides. 

One  hundred  and  twenty-five  pounds  per  square  inch  is 
as  high  a  pressure  per  square  inch  as  should  be  used,  and 
the  most  modern  English  locomotive  practice  takes  forty 
pounds  per  square  inch  as  the  proper  pressure,  the  prac- 
tical result  being  a  great  diminution  in  the  wear  upon  both 
guides  and  slides. 

Let  A  =  the  area  of  a  slide  in  square  inches. 

"    b  =  the  pressure  per  square  inch  allowed. 
Then  will  we  have  for  the  area  of  a  slide, 


Example.  —  Let      P6  =P=40  pounds  per  square  inch. 

"  d  =  32  inches. 

"  N  =  40  per  minute. 

"  L  =48  inches. 
"   (HP)  =  156  horse-power. 
"          b  =  1  25  pounds  per  square  inch. 

"  n  =5. 


OF  THE  STEAM-ENGINE.  49 

Substituting  in  formula  (40),  we  have 


c  39600Q   156 

01  -  -  -  =  6568  pounds. 

48x40 


Substituting  in  formula  (41),  we  have 

6568     KOK 
A  =  -  =  oz.o  square  inches. 


CHAPTER    IV. 

(20.)  Stress  on  and  Dimensions  of  the  Guides.*— 

The  first  requisite  of  a  guide  is  that  it  shall  be  perfectly 
rigid  under  all  circumstances.  .In  many  cases  the  guides 
are  so  attached  to  the  bed-plate  or  frame- work  of  the  engine 
as  to  require  no  calculation  of  their  rigidity. 

Cast  iron  is  used  for  guides  where  they  are  firmly  fastened 
throughout  their  length  to  a  bed-plate  or  framing. 

Under  other  circumstances  wrought  iron  or  steel  is  to  be 
preferred  as  having  greater  moduli  of  elasticity,  and  conse- 
quent rigidity. 

When  it  is  considered  necessary  to  calculate  the  dimen- 
sions of  a  guide,  the  following  method  will  give  a  result 
which  is  safe: 


*  Parallel  motions,  which  take  the  place  of  guides  in  some  engines, 
are  discussed  in  Willis's  Principles  of  Mechanism,  pp.  350-363.  How 
to  Draw  a  Straight  Line,  by  A.  B.  Kempe,  is  a  particularly  interest- 
ing little  book,  giving  all  the  later  discoveries  in  parallel  motions 
which  have  followed  the  invention  of  Peaucellier's  perfect  parallel 
motion. 

5 


50  THE  RELATIVE  PROPORTIONS 

Let  Si  =  the  stress  upon  the  guide  [see  formulae  (39),  (40) 

Art.  (19)]  in  Ibs. 

"     /'  =  the  length  of  guide  in  inches. 
"     JF=the  measure  of  the  moment  of  flexure  of  the 

guide. 
"     E  =  the  modulus  of  f  For  wro't  iron  =  28000000  Ibs. 

elasticity,       1  For  steel          =30000000" 
"     a  =  the  deflection  in  inches. 

We  have  (Weisbach's  Mechanics  of  Engineering,  Sec.  iv., 
Art.  217),  for  a  beam  supported  at  both  ends  and  loaded  in 
the  middle, 

m^L'-SE^  (42) 

48     WE 

Since  perfect  rigidity  is  unattainable,  let  us  concede  a 
deflection,  a  =  ^  of  an  inch,  and  formula  (42)  becomes  for 

wrought  iron, 

--  £T_ 

13440000 
Assuming  a  rectangular  cross-section  for  the  guide, 

Let        b  =  the  breadth  in  inches. 
"          h  =  the  depth  in  inches. 

Then    W=^-, 
12 

and  formula  (43)  becomes 
» 


134400006 

Substituting  the  value  of  $,  derived  from  equation  (39), 
and  extracting  the  cube  root,  (44)  becomes 


(45) 


OF  THE  STEAM-ENGINE.  §\ 

and  substituting  the  value  of  Sl  from  equation  (40), 

k-Mvnr%i£i=*'       (46) 


Example. — Let         t  =  60  inches. 
d  =32  inches. 
L  =48  inches. 
Pb  =P  =  40  Ibs.  per  square  inch. 

6=4  inches. 

"    (HP)  =  156  indicated  horse-powers. 
N  =  40  per  minute. 
n  =  5. 

Using  formula  (45),  we  have 


h  =  0.00889  x  60xp=  =  6.82  inches. 
Mv/25-JL 

Using  formula  (46),  we  have 


3  I  I  K£> 

h  =  0.7071  x  60X  -  -  -  6.82  inches. 

v  4x48x40^/25-1 

(21.)  Distance  between  Guides.  —  It  is  important  to 
know  at  what  angle  of  the  crank  the  connecting-rod  requires 
the  greatest  distance  between  the  guides,  if  the  plane  of 
vibration  of  the  connecting-rod  intersects  them. 

We  can  then  determine  the  least  possible  distance  be- 
tween the  guides,  or  that  position  of  the  connecting-rod 
which,  clearing  all  parts  of  the  engine,  will  make  it  impos- 
sible for  it  to  touch  with  the  crank  at  a  different  angle. 

Solution.  —  (Approximate.) 

Let  r  =  radius  of  crank  =  CB. 
"     l  =  nr  =  length  of  connecting-rod  =  AB. 


We  have  EFiBD::  [2r  -  <1  -  cos  a)]  :  j/F  -  r*  sin2  a. 


52  TEE  RELATIVE  PROPORTIONS 

FIG.  6. 


Let  EF=x,  we  have  BD  =  r  sin  a, 

r2  sin  a  (1  +  cos  a) 
then      x  =  -    v  ^, 

rj/Ti2  —  sin2  a 
and  neglecting  for  the  present  sin"  a  in  the  denominator, 

we  have  x  =  -  sin  a  (1  +  cos  a). 

n 

But    sin    a  =  2  sin  -  cos  —  and  (1  +  cos  a)  =  2  cos2  -,  which 
22  2 

gives  a;  =  —  sin  a  cos3  -.    Differentiating,  we  have 

dx    4r  r         .     a         a  a  i 

—  —  —    —  3  sin2  -  cos2  -  +  cos4  -   , 
da     n  L  22  2J 

and  placing  this  equal  to  0  and  dividing  by  cos2  -, 

2 

3  sin2  ?-  cos2  J 
or    tan2     = 


tan      -!/$-.  578, 

^  =  30°  and  a  =  60°  approximately. 
z 

Showing  that  when  the  crank  forms  an  angle  of  60°  with 
the  centre  line  of  the  cylinder,  we  have  the  maximum  dis- 


OF  THE  STEAM-ENGINE.  53 

tance  from  that  centre  line  required  to  make  the  centre  line 
of  the  connecting-rod  clear  the  guides, 

1.5x0.866          1.3r 

— ____ 

-I/V-. 

To  get  the  whole  distance  between  the  guides  we  multi- 
ply by  2,  giving  &ff=2.6-. 
n 

To  this  value  must  be  added  the  thickness  of  the  connect- 
ing-rod. For  any  point,  as  K,  on  the  connecting-rod,  we 
have,  knowing  the  angle  a  =  60°,  the  proportion, 

AL  :AD::KL:  BD, 
^T 

to  which  we  must  add  the  half  thickness  of  the  connecting- 
rod  and  multiply  by  2  to  get  the  whole  distance  between 
the  guides. 

It  must  be  borne  in  mind,  when  arranging  the  guides, 
that  room  must  be  made  for  the  introduction  of  the  key 
into  the  stub  end  of  the  connecting-rod,  if  that  end  cannot 
be  slid  outside  of  the  guides  at  one  end  of  the  stroke. 

(22.)  The  Connecting-Rod. — The  connecting-rod  of  a 
steam-engine  is  usually  made  from  4  to  8  times  the  length 
of  the  crank — that  is,  2  to  4  times  the  length  of  the  stroke 
of  the  steam-cylinder. 

FIG.  5. 


Referring  to  Fig.  5,  we  see  that  when  the  crank  is  at 
5* 


54  THE  RELATIVE  PROPORTIONS 

right  angles  to  the  centre  line  of  the  piston-rod,  the  strain 
upon  the  connecting-rod  is  a  maximum. 

Let  $  =  the  total  steam -pressure  upon   the  piston-head 

in  pounds. 

"   $2  =  the  stress  upon  the  connecting-rod  in  pounds. 
"     /  =  the  length  of  the  connecting-rod  in  inches. 
"     r  =  the  radius  of  the  crank. 

"    n  =  -  =  the  ratio  of  the  length  of  the  connecting-rod 

to  the  crank. 
We  have    S :  S3 : :  i/n2r*  -  r2 :  nr. 

Therefore,         Sz  =  S—     —.  (47) 

If  in  (47)  we  let  n  =  4,  we  have 

&-fl      4       =1.0328& 

If  we  let  n  =  8,  we  have 

S,  =  S — - — =1.0008& 
t/64-1 

These  numerical  results  show  the  rapidity  with  which  the 
value  of  the  maximum  stress  on  the  connecting-rod  ap- 
proaches the  constant  stress  upon  the  piston-rod  as  the  ratio 
of  the  connecting-rod  to  the  crank  is  increased,  and  further 
by  what  a  small  percentage  =  .03  the  stress  upon  the  con- 
necting-rod is  greater  in  the  extreme  case  for  the  value 
of  n  =  4. 

A  relatively  short  connecting-rod  is  productive  of  econ- 
omy of  material,  and  the  increased  pressure  upon  the  sides 
can  be  provided  for  by  increased  area.  (See  Art.  19.) 

The  connecting-rod,  being  free  to  turn  about  its  pins  at 


OF  THE  STEAM-ENGINE.  55 

either  end,  must  be  regarded  as  a  solid  column  not  fixed  at 
either  end,  but  neither  end  free  to  move  sideways. 

To  determine  the  point  at  which  a  column  of  this  cha- 
racter has  an  equal  tendency  to  rupture  by  crushing  or 
buckling,  we  place  the  formulae  for  crushing  and  buckling 
equal  to  each  other  (Weisbach's  Mechanics  of  Engineering, 
sec.  iv.,  art.  266),  and  letting  d2  =  the  diameter  of  the  con- 
necting-rod, 


or,  substituting  the  values  of  jP  and  W, 

I  \~E 

which  gives  —  =  .7854-*  /— :. 
«j  \  K 

I  /2800000(> 

For  wrought  iron,  —  -  .7854^-^^-  -  23J,  (48) 


and  we  see  that  a  wrought-iron  column  will  rupture  by 
crushing  when  its  diameter  is  greater  than  Jj  of  its  length, 
and  will  rupture  by  buckling  when  its  diameter  is  less  than 
•£%  of  its  length,  and  that  a  steel  column  will  rupture  by 
crushing  when  its  diameter  is  greater  than  y1^  of  its  length, 
and  will  rupture  by  buckling  when  its  diameter  is  less  than 
TV  of  its  length. 

(23.)  Wrought-iron  Connecting  -Rod.—  Let  ^=the 
cross-section  of  the  rod  in  square  inches. 

Let  JfT=the  ultimate  crushing  strength  of  wrought  iron 
per  square  inch. 


56  THE  RELATIVE  PROPORTIONS 

Referring  to  formula  (47),  we  have  for  crushing, 

n  (50) 


Let  P6  =  the  boiler-pressure  in  pounds  per  square  inch. 
"  d2  =  the  diameter  of  the  connecting-rod  in  inches. 
"  d  =  the  diameter  of  the  steam-cylinder  in  inches. 
"  the  factor  of  safety  be  10,  as  before. 


Then  formula  (50)  becomes 

* 

4 


^31000. 


Therefore,  ^  =  0.0179^^x1  -7^7-  (51) 

^  rr—  1 


In  this  formula  for  n  =  4  we  have 


and  as  the  value  of  n  increases  this  quantity  becomes  more 
and  more  nearly  equal  to  unity,  and  can  therefore  be  neg- 
lected in  all  cases  in  which  n  =  4  or  a  greater  number. 

Formula  (51)  then  becomes 

dt  =  0.0179  rfv/F,.  (52) 

Let  (-HP)  =  the  indicated  horse-power. 
"     L       =  the  length  of  the  stroke  in  inches. 
"     N      =  the  number  of  strokes  per  minute. 

Formula  (52)  becomes,  in  terms  of  the  horse-power, 

^  =  12.753^^.  (53) 

The  formula   for  rupture  by  buckling  for   long  solid 


OF  THE  STEAM-ENGINE.  57 

columns,  not  fixed  at  either  end,  is  Weisbach's  Mechanics  of 
Engineering,  sec.  iv.,  art.  266. 

Letting  W=  the  measure  of  the  moment  of  flexure, 

"      E  =  the  modulus  of  elasticity  =  28000000  pounds 
per  square  inch, 

I   =  the  length  of  the  rod  in  inches  =  nr  —  n — , 

(54) 


Substituting  in  this  the  values  of  $  =       n      &   W=  — 


and  .E,  we  have,  with  a  factor  of  safety,  10, 

r  —  2800000°- 
z     64 


n 
Reducing  and  neglecting  the  term  •*/—  -  -  ,  we  have 

<Z2  -  0.02758v/^P6  inches.  (55) 

If  in  this  we  substitute  the  value  of  I  —  n  —  ,  we  have 


d2  -  0.01  95  v'nWET*  inches.  (56) 

If  L-d9  (56)  becomes 

dz  =  0.0195rf|//n5P6  inches.  (57) 

If  we  suppose  the  pressure  of  the  steam  to  be  uniform 
throughout  the  stroke,  formula  (56)  becomes,  in  terms  of  the 
horse-power, 

(58) 


58  THE  RELATIVE  PROPORTIONS 

The  following  rule  may  be  given  for  the  determination  of 
the  diameters  of  round  wrought-iron  connecting-rods : 

Deduce  the  diameter  of  the  connecting-rod  by  the  use  of 
formula  (51)  or  (53),  as  may  be  most  convenient.  Should  this 
diameter  be  less  than  -£%  of  the  length  of  the  rod,  then  use 
formula  (55),  (56),  (57)  or  (58),  as  may  be  most  convenient. 

Example. — Let       L   =  48  inches. 
"          d    =32  inches. 

n    =5. 

"         Pb  =  P=  40  pounds  per  square  inch. 
"          N  =  40  per  minute. 
"    (-HP)  =  156  approximately. 

Substituting  in  formula  (52),  we  have 

d2  =  0.0179  x  32^40  -  3.62  inches, 
or  substituting  in  formula  (53),  we  have 

4  =  12.753    r~T5T~ 

/ =  3.63  inches. 

\48x40 

48 

Since  the  length   of  the   connecting-rod  =  —  x  5  =  120 

2i 

inches,  we  see  that  its  diameter  is  less  than  -fa  of  its  length, 
and  that  we  must  use  formula  (56)  or  (58). 
Substituting  in  (56),  we  have 

dt  =  0.0195]/-25  x  1024  x  2304  x  40  =  4.30  inches. 
Substituting  in  formula  (58),  we  have 

40 


OF  THE  STEAM-ENGINE.  59 

CHAPTER   V. 

(24.)  Steel  Connecting-Rod. — The  same  course  of  rea- 
soning as  that  followed  in  Art.  (13)  gives  us,  for  the  proper 
diameter  of  a  steel  piston-rod  to  resist  safely  a  strain  in 
tension, 

d,  =  0.0105(VT6  inches,  (59) 

which  is  the  same  as  formula  (20),  or 


rf,  -  7.481^|p  inches,  (60) 


LN 

which  is  the  same  as  formula  (21)  for  a  constant  pressure 
of  the  steam  throughout  the  length  of  the  stroke. 

Considering  next  the  strength  of  a.  connecting-rod  to  re- 
sist rupture  by  buckling,  and  letting  .#=42000000  pounds 
per  square  inch  for  steel  in  formula  (54),  substituting  in  a 
similar  manner  to  that  in  Art.  (23)  and  reducing,  we  have 

—  ^42000000. 
I-  64 

8  I       ~ 

Therefore,  neglecting  the  term 


, 
=  0.02492  v4/dWV  (61) 


Substituting  l  =  nr  =  n  —  ,  we  have 
•2 


c/f-  0.0176  lT&Fi                  (62) 
If  in  (62)  we  make 

n  =  4,  or  I  =   2  A  we  have  d,  -  0.0352  ytflfP*  ;  (63) 

n  =  5,  or  I  =  2J  A       "  rf,  =  0.0394  ^UPb  ;  (64) 

n  -  6,  or  J  -   3  A       "  ^2  =  0.0432  ^"OTJ^  ;  (65) 

n  =  7,  or  I  =  3  J  L,       "  ^2  -  0.0466  ^"OTTA  ;  (66) 

w  =  8,  or  I  -   4A       "  ^  -  0.0496  y7^!^  .  (67) 


60  THE  RELATIVE  PROPORTIONS 

81 -^3 

It  must  be  remembered  that  the  term  -i  / — ;=r  is  neglect- 

Vl/w2-l 

ed  in  these  formulae,  and  that  it  should  be  taken  into  consid- 
eration when  n  is  less  than  4. 

If  in  the  above  formulae  we  let  L  =  d,  the  radical  reduces 
to  the  form  d  j/VP6 . 

If  we  consider  the  steam-pressure  uniform  throughout  the 
stroke,  and  substitute  for  P^Ld*  in  (62)  its  value  in  terms 
of  the  horse-power,  we  have 


=  0.469^^^.  (68) 


N 


From  their  experiments  upon  steel,  Kupffer  and  Styffe 
have  arrived  at  the  conclusion  that  the  percentage  of  carbon 
has  no  effect  upon  the  modulus  of  elasticity  of  steel,  and 
give  as  its  value  E=  30000000  per  square  inch. 

In  the  present  discussion  we  have  assumed  the  value  of 
E=  42000000,  as  given  by  Reuleaux,  and  the  ultimate 
strength  to  resist  crushing  ^=100000  pounds  per  square 
inch — a  value  far  less  than  would  be  considered  safe  from 
the  experiments  of  W.  Fairbairn  on  the  mechanical  prop- 
erties of  steel.  (Report  of  the  British  Association  for  1867, 
or  Stoney's  Theory  of  Strains,  vol.  ii.,  page  480.) 

It  is  more  than  probable,  from  a  consideration  of  these 
facts,  that  the  ratio  of  I  to  d*  should  be  much  less  than  16, 
and  require  the  use  in  all  cases  of  formulae  (61)  to  (68), 
inclusive,  for  the  determination  of  the  diameter  of  steel 
connecting-rods. 

Comparing  formula  (22)  for  rupture  by  buckling  of  steel 
piston-rods  with  formulae  (63)  to  (67),  inclusive,  for  rupture 
of  steel  connecting-rod,  we  find  that  when 


OF  THE  STEAM-ENGINE.  61 

(69)  n  =  4,  the  diameter  of  the  connecting-rod  =  1.00  times 

the  diameter  of  the  piston-rod  ; 

(70)  n  =  5,  the  diameter  of  the  connecting-rod  =  1.12  times 

the  diameter  of  the  piston-rod  ; 

(71)  n  =  6,  the  diameter  of  the  connecting-rod  =  1.28  times 

the  diameter  of  the  piston-rod  ; 

(72)  n  =  7,  the  diameter  of  the  connecting-rod  =  1.30  times 

the  diameter  of  the  piston-rod  ; 

(73)  n  =  8,  the  diameter  of  the  connecting-rod  =  1.44  times 

the  diameter  of  the  piston-rod. 

If  the  length  of  the  connecting-rod  be  taken  =  twice  the 
stroke,  the  diameters  of  piston-  and  connecting-rod  are  equal. 

The  following  rule  may  be  given  for  the  determination 
of  the  diameters  of  round  steel  connecting-rods  : 

Deduce  the  diameter  of  the  connecting-rod  from  formula 
(59*)  or  (60),  as  may  be  most  convenient.  If  the  resulting 
diameter  be  less  than  one-sixteenth  of  the  length  of  the  rod, 
then  use  some  one  of  the  formulas  (61)  to  (73*),  inclusive. 

Example.  —  Let        L  =48  inches. 

"  d  =32  inches. 

n  =  5  inches. 

P6  =  P=40  pounds  per  square  inch. 
.  "         N  =  40  per  minute. 
"    (.HP)  =  156. 

Substituting  these  values  in  formula  (59),  we  have 

d2  =  0.0105  x  32  1/40  =  2.12  inches, 
or,  substituting  in  formula  (6.0),  we  have 


d,  =  7.481     --  =  2.13  inches. 
\48x40 

•jJg-  of  24x5  =  120  =  7^  inches,  and  we  see  that  we  must 
use  one  formula,  (61)  to  (73). 


62  THE  RELATIVE  PROPORTIONS 

Substituting  in  formula  (64),  we  have  for  n  =  5 

d.2  =  0.0394  y'  1024x2304^40  =  3.88  inches ; 
or,  substituting  in  formula  (68),  we  have 

c?2  =  0.469 *!—  -  =  3.89  inches. 

Referring  to  example  for  Art.  13,  we  find  the  diameter  of 
a  steel  piston-rod  to  be  dl  =  3.47  inches,  and  substituting  in 
formula  (70),  we  have 

d,  =  1.124  =  3.47  x  1.12  =  3.89  inches. 

(25.)  General   Remarks   concerning  Connecting- 

Rods. — The  discussion  of  Art.  (14)  will  apply  with  little 
modification  to  connecting-rods.* 

If  we  take  formula  (27),  as  before,  for  the  volume  of  a 
connecting-rod,  we  have 

"-(?)  (¥)• 

Substituting  in  this  the  value  of  d2  from  equation  (68), 
we  have,  letting  C=  constant, 


Reducing  V,  =  W^^-.  (76) 

Equation  (76)  shows  that  the  volume  of  the  connecting- 
rod  is  affected  by  the  varying  conditions  in  a  similar  man- 
ner to  a  piston-rod,  excepting  that  the  square  of  the  ratio  n 
enters  in  and  affects  the  volume  of  the  rod. 

*  It  is  customary  to  make  round  connecting-rods  with  a  taper  of 
about  one-eighth  of  an  inch  per  foot  from  the  centre  to  the  necks, 
which  should  be  of  the  calculated  diameter.  Experiment  does  not 
show  an  increased  strength  from  a  tapering  form. 


OF  THE  STEAM-ENGINE.  63 

Thus,  if  we  first  take  n  =  4,  and  then  n  =  8,  we  see  that 
the  volume  in  the  first  case  is  but  J  of  the  volume  in  the 
latter  case. 

The  increase  of  the  area  of  the  slides  to  provide  for 
shorter  connecting-rods  is  quite  slow  (see  Art.  19).  It  is 
customary  to  make  connecting  and  parallel  rods  of  a  rectan- 
gular cross-section  in  locomotive  practice.  When  this  is 
done  it  will  be  safe  to  make  the  smaller  of  the  rectangular 
dimensions  equal  to  the  diameter  of  a  round  rod  suitable 
to  withstand  the  strains  to  which  it  will  be  subjected.  The 
resistance  of  a  long  rectangular  column  to  rupture  varies  as 
the  cube  of  its  smallest  dimension  of  cross-section.  (See 
Weisbach's  Mechanics  of  Engineering,  sec.  iv.,  art.  266.) 

(26.)  Connecting-Rod  Straps. — By  means  of  gibs  and 
keys  the  straps  draw  the  brasses  solidly  against  the  stub-end 
of  the  connecting-rod. 

If  necessary  the  uniform  cross-section  of  the  straps  can 
be  preserved  by  thickening  them-  at  the  point  where  they 
are  slotted  to  receive  the  gib  and  key.  (See  Art.  15.) 

Fig.  7  a  shows  the  usual  form  of  strap  for  locomotives, 
and  Fig.  7  b  a  solid  stub-end  in  which  the  keys  are  used 
only  to  set  the  brasses  without  moving  the  strap.  Fig.  7  c 
represents  the  ordinary  form  of  strap  in  which  the  brasses 
and  strap  are  held  by  means  of  the  gib  and  key. 

(27.)  Wrought-Iron  Strap.— 

Let  FI  =  the  area  of  one  leg  of  the  strap  in  square  inches. 
"       the  safe  strain  per  square  inch  =  5000  pounds. 
"    d  =  the  diameter  of  the  steam-cylinder  in  inches. 
"    Pb  =  the  pressure  per  square  inch  of  the  steam. 

We  have    2  x  5000^  =  0.7854P6d2 ; 

therefore  Fl  =  0.000078P6d2  square  inches.    (77) 


64  THE  RELATIVE  PROPORTIONS 

or  if  we  assume  the  pressure  to  be  uniform,  and 

Let         L  =  the  length  of  the  stroke  in  inches, 
.Af=the  number  of  strokes  per  minute, 
"    (.HP)  =  the  horse-power  (indicated), 

FI  =  39.6^^  square  inches.  (78) 

LN 


OF  THE  STEAM-ENGINE.  65 

Example. — Let         d  =  32  inches. 
L  =  48  inches. 
N=  40  per  minute. 
P6  =  P  =  40  pounds  per  square  inch. 
"  (HP)  =156  indicated  horse-powers. 

Substituting  in  formula  (77),  we  have 

Fl  -  0.000078  x  40  x  1024  =  3.19  square  inches, 
or  substituting  in  formula  (78),  we  have 

F,  =  39.6    156    =  3.21  square  inches. 
48  x  40 

(28.)  Steel  Strap. — Let  9000  pounds  per  square  inch 
equal  the  safe  working-strain  in  tension. 

Let         Fl  =  the  area  of  one  leg  of  the  strap  in  sq.  inches. 
d  =  the  diameter  of  the  steam-cylinder  in  inches. 
Pb  =-  the  steam-pressure  per  square  inch. 
L  =  the  length  of  stroke  in  inches. 
JV=the  number  of  strokes  per  minute. 
"    (-HP)  =  the  indicated  horse-power. 

We  have,  as  in  the  preceding  article, 

2x9000^  =  0.7854^, 
therefore  Px  =  0.0000437d2P6  square  inches,    (79) 

or  assuming  the  stream-pressure  to  be  uniform  throughout 
the  stroke,  and  substituting  for  d?Pb  its  value  in  terms  of 
the  horse-power, 

F,  =  22.034^2  square  inches.  (80) 

Dividing  formula  (80)  by  formula  (78),  we  find  that  the 
cross-section  of  a  steel  strap  is  but  -J  of  the  cross-section  of 

a  wrought-iron  strap  of  equal  strength. 
6* 


66  THE  RELATIVE  PROPORTIONS 

Example. — Let  the  data  be  the  same  as  in  the  example 
appended  to  the  preceding  article. 
Substituting  in  formula  (79),  we  have 

Jl  =  0.0000437  x  1024  x  40  =  1.77  square  inches. 

Substituting  in  formula  (80),  we  have 
"i  r^^? 

Fx  -  22.034— -4-  - 1.78  square  inches. 
48x40 


CHAPTER   VI. 

(29.)  The  Crank-Pin  and  Boxes.— The  crank-pin  has 
ever  been  one  of  the  most  troublesome  parts  of  the  steam- 
engine  to  the  mechanical  engineer.  The  mere  determination 
of  its  proportions,  so  that  it  will  not  break  under  the  strain 
put  upon  it  by  the  pressure  of  the  steam  upon  the  piston- 
head,  does  not  suffice,  and  often  results  in  trouble  from 
heating  when  the  engine  is  at  work.  It  therefore  becomes 
a  first  consideration  to  so  proportion  crank-pins  as  to  pre- 
vent heating,  their  strength  being  a  matter  of  secondary 
importance,  to  be  afterward  investigated  if  it  is  deemed 
necessary  to  do  it. 

Before  taking  up  the  mathematical  part  of  our  consid- 
eration, it  will  be  of  practical  value  to  quote,  from  the 
writings  of  General  Morin,  the  following  remarks : 

"  But  it  is  proper  to  observe  that  from  the  form  itself  of 
the  rubbing  body  (cylindrical)  the  pressure  is  exerted  upon 
a  less  extent  of  surface  according  to  the  smallness  of  the 
diameter  of  the  journal,  and  that  unguents  are  more  easily 
expelled  with  small  than  with  large  journals.  This  circum- 
stance has  a  great  influence  upon  the  intensity  of  friction, 
and  upon  the  value  of  its  ratio  to  the  pressure. 


OF  THE  STEAM-ENGINE.  67 

"  The  motion  of  rotation  tends  of  itself  to  expel  certain 
unguents  and  to  bring  the  surfaces  to  a  simply  unctuous 
state.  The  old  mode  of  greasing,  still  used  in  many  cases, 
consisted  simply  in  turning  on  the  oil  or  spreading  the  lard 
or  tallow  upon  the  surface  of  the  rubbing,  and  in  renewing 
the  operation  several  times  in  a  day. 

"  We  may  thus,  with  care,  prevent  the  rapid  wear  of 
journals  and  their  boxes  ;  but  with  an  imperfect  renewal 
of  the  unguent,  the  friction  may  attain  .07,  .08,  or  even  .10, 
of  the  pressure. 

"  If,  on  the  other  hand,  we  use  contrivances  which  renew 
the  unguent,  without  cessation,  in  sufficient  quantities,  the 
rubbing  surfaces  are  maintained  in  a  perfect  and  constant 
state  of  lubrication,  and  the  friction  falls  as  low  as  .05  or 
.03  of  the  pressure,  and  probably  still  lower. 

"  The  polished  surfaces  operated  in  these  favorable  con- 
ditions became  more  and  more  perfect,  and  it  is  not  sur- 
prising that  th'e  friction  should  fall  far  below  the  limits 
above  indicated."  (Bennett's  Morin,  pp.  307,  308.) 

If  the  unguents  are  expelled  by  extreme  pressure,  so  that 
the  surfaces  are  simply  unctuous,  the  friction  increases  rap- 
idly, and  the  surfaces  begin  to  heat  and  wear  immediately. 

These  statements  apply  with  equal  force  to  cast  iron  and 
cast  iron ;  cast  iron  and  wrought  iron ;  cast  iron  and  brass 
or  Babbitt's  metal;  or  with  steel  or  wrought  iron  in  the 
place  of  cast  iron. 

The  supposed  superiority  of  brass  or  Babbitt's  metal  lined 
boxes  over  iron  boxes  in  positions  very  liable  to  heating 
lies  in  their  greater  softness  and  conductivity  for  heat. 
Brass  will  conduct  heat  away  from  two  to  four  times  as 
rapidly  as  iron.  However,  the  film  of  unguent  interposed 
may  render  the  conductivity  of  brass  of  less  avail  than  is 
generally  supposed,  and  the  advantage  lies  only  in  the  fact 
that,  being  a  softer  metal,  in  case  of  heating,  the  surface  of 


68  THE  RELATIVE  PROPORTIONS 

the  softer  metal  receives  the  principal  damage.  Phosphor 
bronze,  which  is  a  patented  alloy,  being  nothing  more  than 
brass  or  gun-metal  in  which  the  formation  of  oxide  has  been 
prevented  by  the  introduction  of  phosphorus,  is  coming  into 
general  use  for  positions  in  which  the  wear  is  very  great. 

It  is,  perhaps,  good  practice  to  use  brass  or  soft  metal 
wherever  the  pressure  exceeds  125  pounds  per  square  inch 
of  projected  area.  At  lower  pressures  a  good  lubricating  oil 
may  be  relied  upon  to  form  a  film  and  run  without  breaking 
at  ordinary  speeds.  (The  continuity  of  the  film  of  lubricant 
is  affected  by  so  many  different  conditions  that  it  is  impos- 
sible to  fix  any  exact  limit  of  pressure.) 

With  soft  metal  or  brass  bearings  good  results  can  be  ob- 
tained at  pressures  of  1000  pounds  or  more  per  square  inch 
of  projected  area.  (See  Hand-Book  of  the  Steam-Engine, 
Bourne,  page  183,  where  1400  pounds  per  square  inch  is 
given  as  the  greatest  pressure  per  square  inch  of  projected 
area  allowable  on  crank-pins.  Arthur  Kigg,  A  Practical 
Treatise  on  the  Steam-Engine,  page  147.)  If,  however,  the 
film  of  unguent  does  break  at  these  higher  pressures,  heat- 
ing begins  almost  instantly ;  and  if  the  surfaces  in  contact 
are  both  of  hard  metal,  as  iron  and  iron,  injury  to  both  at 
once  results,  while,  if  the  boxes  are  brass  or  some  of  the 
softer  metals,  the  continuity  of  the  surface  film  may  be  re- 
stored by  increased  lubrication  or  by  stopping  and  cooling 
as  soon  as  heating  is  observed. 

Several  expedients  are  used  to  keep  bearings  which  have 
a  tendency  to  heat  cool  until  they  have  worn  smooth. 

The  introduction  of  rotten  stone  or  sulphur  with  oil  is 
perhaps  the  best.  Quicksilver  or  lead-filings,  introduced 
with  oil,  coat  the  rubbing  surfaces  and  diminish  the  heat- 
ing where  the  rubbing  surfaces  are  very  much  scored.  (See 
The  Working  Engineer's  Practical  Guide,  pages  48  and  49, 
Joseph  Hopkinson.) 


OF  THE  STEAM-ENGINE. 


69 


A  great  increase  of  the  velocity  of  the  rubbing  surfaces 
renders  bearings  more  liable  to  heat  than  a  great  increase 
in  pressure,  although  the  total  amount  of  work  done  by 
friction  is  the  same  in  both  cases,  and  is  probably  accounted 
for  by  the  more  rapid  expulsion  of  the  lubricant. 

As  the  cause  of  the  heating  of  bearings,  when  they  are  of 
tolerably  good  workmanship,  is  the  transformation  of  the 
work  of  friction  into  heat,  we  see  that  it  is  necessary  to 
reduce  the  friction  as  much  as  possible  by  the  perfect 
smoothness  of  surfaces  in  contact,  the  interposition  of  lubri- 
cants, and  the  reduction  of  the  speed  and  pressure  upon  the 
rubbing  surfaces. 

In  all  machines  there  is  a  limit  below  which  we  cannot 
reduce  the  speed  and  pressure  of  the  rubbing  surfaces,  and 
•we  must,  therefore,  so  proportion  journal-bearings  as  to 
cause  no  more  work  due  to  friction — i.  e.,  heat — to  be  pro- 
duced than  can  be  conveyed  away  by  the  unguents,  the 
atmosphere  and  the  conductivity  of  the  metals  without 
raising  the  temperature  of  the  bearing  appreciably. 

From  the  statistics  of  the  working  of  the  crank-pins  of 
four  screw  propellers  in  the  United  States  Navy  (Van 
Buren,  Strength  of  Iron  Parts  of  Steam  Machinery,  page  24) 
we  take  the  following  statement  and  table : 

"  The  crank-pins  of  these  vessels  worked  cool,  giving  but 
little  trouble,  which  is  the  ex- 
ception rather  than  the  rule  for 
screw-engines." 

The  projected  area  of  crank- 
pin  journal,  given  in  column  7, 
is  that  rectangular  area  formed 
by  a  central  section  of  the  crank-pin  journal  in  the  direc- 
tion of  its  length.     Shown  cross-hatched  in  Fig.  8. 

Columns  1,  2,  4,  5  and  6  are  given.  Columns  3,  7,  8,  9, 
10  and  11  are  calculated  from  them. 


FIG.  8. 


70 


THE  RELATIVE  PROPORTIONS 


In  calculating  column  9  from  columns  3  and  8  we  have 
assumed  the  coefficient  of  friction  at  .05,  which  is  the  high- 
est value  given  by  General  Morin  for  constant  lubrication, 
and  probably  greater  in  the  present  cases.* 

Column  11  is  derived  from  columns  3  and  7.  Column 
10  is  derived  from  columns  7  and  9. 

TABLE  HI. 


1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

| 

2 

11 

1 

d 

Q, 

•s 

t. 
2.2 

j. 

Hi 

°  .• 

a 

a 

%£ 

1 

P. 

»1 

s  S 

C.-C  S 

If 

1 

t 

I 

0 

a 

^ 

u 

S| 

§f? 

NAME  OP  VESSEL. 

Diameter  of 
inder. 

Pressure  per 
inch. 

h 
111 

Numbur  of  st 
minute. 

Length  of  era 

1 

IP  rejected  aret 
pin. 

11 

Total  work  of 
foot-pounds 

Work  of  frict 
inch  of  proj 
in  foot-pound 

i 

2§, 

in. 

Ibs. 

Ibs. 

in. 

in. 

sq.  in. 

Swatara.  

36 

40 

40716 

160 

12 

8.5 

102. 

178. 

362372 

3552.6 

399.5 

Saco  

30 

40 

28274 

180 

9 

75 

67.5 

176.7 

249801 

3700.7 

419. 

Wampanoag... 

100 

40 

314160 

62 

27 

16. 

432. 

129.8 

2038898 

4719.7 

727.2 

Wabash  

72 

28 

114002 

100 

16 

15. 

240. 

196.3 

1118910 

4662.1 

475. 

Avera 

4159. 

505. 

From  column  10  of  the  table  we  find  the  average  amount 
of  work  per  square  inch  of  projected  area  of  crank-pin 
journal,  which,  in  the  cases  cited,  has  been  borne  without 
heating,  to  be  4159  foot-pounds  per  minute;  and  in  making 
use  of  this  quantity  in  our  subsequent  calculations,  we  are 
on  the  safe  side  if  the  coefficient  of  friction  (assumed  at  .05) 
has  not  been  taken  too  small. 

(30.)  The  Length  of  Crank-Pins.— Let  d  =  the  diam- 
eter of  the  piston-head  in  inches,  P=the  mean  pressure  in 
the  steam-cylinder  in  pounds  per  square  inch. 

*  It  is  probable  that  the  coefficient  of  friction,  for  crank-pins  of 
marine  propellor-engines  under  ordinary  conditions,  is  9  or  10  times 
greater  than  the  assumed  0.05. 


OF  THE  STEAM-ENGINE.  71 

Then  7854cPP  =  the  mean  pressure  on  the  piston-head  in 
pounds. 

Let  /  =  the  coefficient  of  friction. 
"    4  =  the  length  of  the  crank-pin  journal  in  inches. 
"   da  =  the  diameter  of  the  crank-pin  journal  in  inches. 

The  mean  force  of  friction  at  the  rubbing  surfaces  of  any 
crank-pin  journal  per  square  inch  of  projected  area  is 


Let  jY=the  number  of  strokes  per  minute  (equal  twice 

the  number  of  revolutions). 
"   w  =  the  work  of  friction  per  minute. 
The  space  passed  over  by  the  force  due  to  friction  in  one 
minute 

1.5708  Nd3  inches, 


and  we  have  for  the  work  of  friction  —  i.  e.}  heat  —  per  minute, 


w  =  1.5708  x  .7854/-  inch-pounds. 
4 

From  this  formula  the  diameter  of  the  crank-pin  journal 
has  vanished.  Why  it  has  vanished  will  be  understood 
when  we  observe  that  the  force  per  square  inch  of  project- 
ed area  due  to  friction  is  inversely  as  the  diameter  of  the 
journal  ;  while  the  space  passed  over  by  this  force  is  directly 
as  its  diameter. 

Replacing  w  in  the  last  formula  by  the  mean  value  de- 
rived from  column  10  of  the  table,  equal  4159  foot-pounds 
equal  49908  inch-pounds,  we  have 

49908  =  1.2337/^^2. 
4 

Therefore,      4  =  .0000247  /P^cf  =  12.454/^^.  (81) 


72  THE  RELATIVE  PROPORTIONS 

Considering  formula  (81),  we  see  that  the  length  of  the 
crank-pin  increases  and  decreases  with  the  coefficient  of 
friction,  the  mean  steam-pressure  per  square  inch  the  num- 
ber of  strokes  per  minute,  and  with  the  square  of  the  diam- 
eter of  the  steam-cylinder. 

A  consideration  of  the  component  formulae  of  (81)  shows 
that  as  the  crank- pin  journal  decreases  in  size  the  pressure 
per  square  inch  becomes  greater ;  but  if  this  reduction  in 
size  is  obtained  by  a  diminution  of  the  diameter  (ds)  of  the 
crank-pin  journal,  the  work  per  square  inch  of  projected 
area  is  not  increased,  for  the  velocity  of  the  rubbing  sur- 
faces by  this  means  is  decreased  in  the  same  ratio  as  the 
pressure  is  increased. 

Within  reasonable  limits  as  to  pressure  and  speed  of  rub- 
bing surfaces,  the  general  law  may  be  enunciated : 

The  longer  any  bearing  which  has  a  given  number  of 
revolutions  and  a  given  pressure  to  sustain  is  made,  the 
cooler  it  will  work,  and  its  diameter  has  no  effect  upon  its 
heating. 

Example.— Letd  =  30",  JV=180,  and  P=40  pounds  per 
square  inch.  We  have,  by  substitution,  in  formula  (81), 

/3  -  .0000247  x/x  40  x  180  x  900  - 160. /. 
If  in  this  we  take/=.03  to  .05  for  perfect  lubrication, 
we  have    4  =  4.8"  to  8". 

If  we  take/=.08  to  .10  for  imperfect  lubrication, 
we  have    4  =  12.8"  to  16". 

The  results  show  the  great  advantages  arising  from  con- 
stant oiling  of  bearings  and  smoothness  of  surfaces. 

NOTE. — If  0.05  be  taken  as  the  coefficient  of  the  force  of  friction, 
we  obtain  the  average  length  of  the  crank-pins  quoted  in  Table  III., 


OF  THE  STEAM-ENGINE.  73 

Art.  (29).  About  one-quarter  of  the  length  required  for  propellor 
crank-pins  will  serve  for  the  pins  of  side-wheel  engines  with  good 
results,  and  one-tenth  for  locomotive  or  stationary  engines. 

(31.)  Locomotive  Crank-Pins,  Length  and  Diameter. 

—  If  for  locomotive  crank-pin  journals  we  assume  N,  the 
number  of  strokes  per  minute  =  600. 

P  the  pressure  per  square  inch  in  pounds  =  150,  the  form- 
ula (81)  being  changed  to 


by  removing  the  decimal  point  one  place  to  the  left,  will 
give  the  length  of  journal  commonly  assumed  in  successful 
practice,  if  we  assume  the  coefficient  of  friction  at  .06. 
The  above  formula  then  becomes 

4  =  .013d2.  (82) 

This  formula  would  prove  the  amount  of  heat  per  square 
inch  of  projected  area  conveyed  away  from  the  crank-pins 
of  locomotives  to  be  ten  times  greater  than  in  the  case  of 
marine  engines,  did  not  the  variations  of  speed  and  frequent 
stoppages  of  a  locomotive  prevent  comparison. 

Example.  —  Let  d  =  18  inches.     We  have 
It  =  .013x324  =  4.21  inches. 

The  diameters  of  locomotive  crank-pins  are  usually  taken 
equal  to  their  length. 
7 


74  THE  RELATIVE  PROPORTIONS 


CHAPTER   VII. 

(32.)  Diameter  of  a  Wrought-Iron  Crank-Pin  for  a 
Single  Crank.*  —  It  is  necessary,  first,  to  determine  its 
length  by  formula  (81),  and  with  this  length  to  determine 
the  proper  diameter. 

Let    a  =  the  deflection  of  pin  under  stress  in  inches. 
"     $  =  the  stress  on  pin  in  pounds. 
"    E  =  the  modulus  of  elasticity  of  wrought  iron  =  28000000 

pounds. 

"    JF=  the  measure  of  the  moment  of  flexure  of  the  pin. 
"     13  =  the  length  of  journal  in  inches. 

The  deflection  of  a  beam  fixed  at.  one  end  and  loaded  at 
the  other  (Weisbach's  Mechanics  of  Engineering  y  sec.  iv., 
art.  217)  is 

SP 

ZWE 

If,  again,  the  beam  be  supposed  to  be  uniformly  loaded 
and  fixed  at  one  end,  its  deflection  will  be  (Weisbach's 
Mechanics  of  Engineering  ,  sec.  iv.,  art.  223) 


_ 
= 


and  if  for  the  load  at  the  end  we  concede  a  deflection  of 
•j-J-g-  of  an  inch,  we  have  for  the  same  load  under  the  two 
cases  above  mentioned 

<Zi  =  .01  inch, 
«2  =  .0038  inch. 

*  In  Arts.  54  and  55  will  be  found  a  discussion  of  the  stresses  on 
crank-pins  for  double  and  triple  cranks. 


OF  THE  STEAM-ENGINE.  75 

Then,  taking  the  most  unfavorable  case — i.  e.,  the  load  at 
the  end — we  have, 

TT  d2 
letting  S= P6, 

letting  P6  =  the  greatest  pressure  of  steam  in  cylinder  equal 
the  boiler-pressure, 

letting  W-^' 

o4 

"nnr =  "g^TTI 


64 


,  (84) 
for  a  constant  steam-pressure. 

Example.  —  Let  P6  =  60  pounds  per  square  inch. 

"     /3  =  8  inches. 

"     d  =  30  inches. 
Substituting  in  formula  (84),  we  have 

d3  =  .066|/'  60x51  2x900  =  4.79  inches. 

There  is  no  need  of  an  investigation  of  the  strength  of  a 
crank-pin,  as  the  condition  of  rigidity  gives  a  great  excess 
of  strength. 

(33.)  Steel  Crank-Pins.—  The  length  of  a  steel  crank- 
pin  is  just  the  same  as  that  of  a  wrought-iron  pin,  and  the 
modulus  of  elasticity  of  steel  is  so  nearly  equal  to  that  of 
wrought  iron  as  to  make  formula  (84)  serviceable  for  both 
steel  and  wrought  iron  alike. 


76  THE  RELATIVE  PROPORTIONS 

The  advantages  of  steel  crank-pins  over  wrought  iron  are 
their  greater  strength,  and  the  possibility  of  obtaining  a 
much  smoother  surface  because  of  the  homogeneous  struc- 
ture of  steel.  Their  disadvantage  is  their  liability  to  sud- 
den fracture  when  not  working  truly  for  any  reason,  as  inac- 
curacy of  workmanship  or  wrenching  as  in  a  marine-engine. 

(34.)  Diameters  of  Crank-Pins  from  a  Considera- 
tion of  the  Pressure  upon  them.  —  Keferring  to  the  table, 
we  find  the  average  pressure  per  square  inch  of  projected 
area  to  be  about  500  pounds. 

If  now  we  divide  the  whole  pressure  upon  the  crank-pin 
by  500,  we  obtain  the  projected  area  required  when  this 
limit  is  not  to  be  exceeded. 


The  equation     tt  —     --  gives  d3  -  .00157—.  (85) 
4  x  out)  '3 

Example.—  Let  P=  40  pounds,   4  =  8",   d  =  30", 

d,  =  .00157  9QOQX4°  -  7.06  inches. 
8 

This  latter  method  is  perhaps  the  most  practical,  and 
has  the  advantage  of  limiting  the  pressure.  It  will  almost 
always  give  larger  results  than  the  preceding  method. 

If  we  assume  the  pressure  to  be  uniform  throughout,  the 
stroke  (85)  becomes,  in  terms  of  the  horse-power, 

(86) 


(35.)  Of  the  Action  of  the  Weight  and  Velocity  of 
the  Reciprocating  Parts.—  By  the  reciprocating  parts 
we  mean  the  piston-head,  piston-rod,  cross-head  or  motion- 
block,  and  the  connecting-rod  ;  also,  in  a  vertical  engine,  the 
working-beam  if  one  is  used. 


OF  THE  STEAM-ENGINE. 


77 


We  are  obliged  to  neglect  the  action  of  friction  from 
the  impossibility  of  determining  it,  and  we  will  also  at  first 
neglect  the  influence  of  gravity  and  the  angular  position  of 
the  connecting-rod — i.  e.,  suppose  it  to  be  of  infinite  length. 


In  order  to  clearly  comprehend  the  motion  of  the  recip- 
rocating parts  in  a  horizontal  direction  for  a  horizontal  en- 
gine, Fig.  9  (A),  lay  down  a  horizontal  line,  O  X,  and 
divide  it  into  12  equal  spaces,  O,  1,  2,  3,  etc.,  to  12;  at  these 
points  draw  ordinates  at  right  angles  to  O  X. 

With  any  centre  upon  the  line  O  X,  as  2,  and  any  radius 
as  2  C,  describe  the  circle  O  C  B  4  D  O,  and  beginning  at  O 
divide  the  circumference  of  this  circle  likewise  into  12  equal 
parts. 

Let  2  C  represent  the  position  of  the  centre  line  of  the 
crank,  let  2  be  the  centre  of  the  crank-shaft,  and  let  C  be 
the  centre  of  the  crank-pin. 

Let  the  angle  a  =  C  2  O  be  the  variable  angle  formed  by 
the  centre  line  of  the  crank  with  the  horizontal  O  X. 

r* 


78  THE  RELATIVE  PROPORTIONS 

Let   r  =  the  radius  of  the  crank  20. 
"     $=the  space  passed  over  by  the  piston-head  (neces- 

sarily in  a  horizontal    direction  OX)  in  the 

time  t. 
"    V=*  the  angular  velocity  of  revolution  of  the  crank  (as- 

sumed constant). 
"    T=the  time  of  one  revolution  of  the  crank. 

We  have  F=—  , 

£  =  r(l-cosa).  (87) 

Differentiating  (87),  we  have 

dS=rsmada.  (88) 

Letting  v  represent  the  velocity  of  the  piston-head  in  a 
horizontal  direction,  and  dividing  (88)  by  dt,  we  have 

f  -'-»•*• 

dS  ,   da  2n 

and  since  —  =  v  and  —  —  y=  —  , 


(90) 


Differentiating  equation  (90),  we  have 


-  27TT  x/v^s. 

dv  = COS  a  d  a,  (91) 

and  dividing  by  dt,  as  before,  we  have  for  the  acceleration 

dv^_2nr          ^?«./??V         a  (92^) 

dt~  T         "dt     \  T  i 

If  we  assume  the  angular  velocity  =  F=  —  =  1,  we  can 
graphically  compare  the  curves  of  these  equations,  Fig.  9  (A). 


OF  THE  STEAM-ENGINE.  79 

The  ordinates  to  the  curve  O  B  E  F  12  show  the  distance 
of  the  piston-head  from  its  starting-point  during  one  revo- 
lution, which  can  be  calculated  also  from  equation  (87). 
The  ordinates  to  the  curve  O  B  6  G  12  show  the  velocities 
of  the  piston-head  during  one  revolution,  which  can  also  be 
calculated  from  equation  (90).  The  ordinates  to  the  curve 
H  3  K  9  L  show  the  accelerations  of  the  velocity  of  the 
piston-head  during  one  revolution,  which  can  also  be  calcu- 
lated from  equation  (92).  All  of  the  reciprocating  parts 
are  supposed  to  move  in  conjunction  with  the  piston-head. 

If  now  we  wish  to  determine  the  acceleration  of  the  pis- 
ton-head for  every  position,  Fig.  9  (B),  on  the  line  0  X,  we 
lay  off  from  O  toward  X  the  ordinates  to  the  curve  of  dis- 
tances for  six  points,  and  at  these  points  erect  ordinates 
taken  from  the  same  positions  and  equal  to  the  ordinates  to 
the  curve  of  accelerations.  The  extremities  of  these  ordi- 
nates can  be  joined  by  a  straight  line,  A  B.  For,  if  we  sub- 
stitute in  equation  (87)  the  value  of  cos  a  derived  from 

equation  (92)  we  have,  letting  y  =  — —  =  the  acceleration, 

at 


Therefore  S=r->  (93) 


which  is  the  equation  of  a  straight  line  cutting  O  X  at  a 
distance  r  from  the  origin. 

Referring  to  equation  (92),  we  observe  that  the  accelera- 
tion varies  with  the  cosine  of  a,  and  therefore  is  a  maxi- 
mum for  cos  a  =  1.  This  gives 

jr-FV,  (94) 


80  THE  RELATIVE  PROPORTIONS 

which  is  the  expression  for  the  acceleration  due  to  centrif- 
ugal force.  If,  therefore,  we  wish  to  balance  a  horizontal 
engine  at  its  dead  points,  we  must  use  a  counter- weight  so 
placed  that  its  statical  moment  is  equal  to  the  statical  mo- 
ment of  the  reciprocating  parts  supposed  to  be  concentrated 
at  the  centre  of  the  crank-pin.  The  engine  cannot  be  bal- 
anced for  any  other  than  its  dead  points :  and  when  the 
crank  is  at  right  angles  to  the  centre  line  of  the  cylinder, 
nearly  the  full  centrifugal  force  of  the  counter-weight  is  felt. 
In  engines  driven  at  widely  different  speeds — as,  for  in- 
stance, a  locomotive — the  use  for  counter-weights  seems  to 
be  the  only  practical  method,  and  therefore  the  recipro- 
cating parts  should  be  made  as  light  as  is  consistent  with 
sufficient  strength  to  resist  the  stresses  coming  upon 
them. 

In  the  case  of  engines  running  at  a  constant  speed,  the 
weight  and  velocity  of  the  reciprocating  parts  affect  the 
stress  upon  the  crank-pin  in  a  manner  which  can  be  deter- 
mined, and  are  therefore  worthy  of  consideration. 

Referring  to  Fig.  9  (B),  we  see  that  the  piston  resists, 
leaving  each  end  of  the  steam-cylinder  with  a  force  equal 
to  its  centrifugal  force  (equation  94), 
Fio.9(B).  and  further,  that  the  intensity  of 

this  force  diminishes  uniformly  from 
the  end  to  the  centre  of  stroke  3, 
where  it  is  zero.  It  will,  therefore, 
be  at  once  recognized  that  an 
amount  of  work  represented  by  the 
area  of  the  triangle  A3O  is  sub- 
tracted from  the  work  impressed 

upon  the  piston  by  the  steam  during  the  first  half  stroke, 
and  that  an  equal  amount,  represented  by  the  area  of  the 
triangle  3  X  B,  is  added  to  the  work  impressed  upon  the 
piston  during  the  last  half  of  the  stroke. 


OF  THE  STEAM-ENGINE. 


81 


In  an  ordinary  indicator  diagram,  Fig.  10,  we  have  the 
means  of  measuring  the  force  acting  upon  the  piston-head 


at  every  point  of  the  stroke.  (For  a  thorough  discussion 
and  explanation  of  the  steam-engine  indicator  refer  to  The 
Richards  Steam-engine  Indicator,  Porter,  or  The  Engine- 
Room,  and  who  should  be  in  it.) 

Let  the  line  O  X  represent  the  atmospheric  line  of  an 
indicator  diagram  taken  from  a  non-condensing  engine  cut- 
ting off  at  one-half  stroke. 

Let  Pb  —  the  initial  pressure  of  the  steam  upon  the  piston- 
head  in  pounds  per  square  inch. 

"  Pf  =  the  final  pressure  of  the  steam  upon  the  piston-head 
in  pounds  per  square  inch. 

"  y  =  the  resistance  due  to  the  inertia  of  the  reciprocat- 
ing parts  in  pounds  per  square  inch. 

If  now  we  impose  the  condition  that  the  initial  and  final 
pressures  upon  the  crank-pin  be  equal,  we  must  have 


Therefore 


y 


(95) 
(96) 


Let  A  =  the  area  of  the  piston-head  in  square  inches. 
"    G  =  the  weight  of  the  reciprocating  parts. 


82  THE  RELATIVE  PROPORTIONS 

We  have,  equating  equations  (94)  and  (96), 


Transposing  and  substituting  the  values, 

A  =  0.7854cP  square  inches, 

d  =  diameter  of  steam-cylinder  in  inches, 

g  =  32.2  feet  per  second, 
JV=the  number  of  strokes  per  minute, 

F—  —  N  feet  per  second, 
60 

r  =  the  radius  of  the  crank  in  feet, 
we  have 


(98) 


Considering  formula  (98),  we  see  that  if  the  initial  and 
final  steam-pressure  Pb  and  Pf  become  nearly  or  quite  equal, 
the  weight  of  the  reciprocating  parts  should  be  nothing. 
Of  course,  the  only  method  in  such  a  case  is  to  make  the 
reciprocating  parts  as  light  as  possible.  We  further  ob- 
serve that  the  weight  of  the  reciprocating  parts  increases  as 
the  difference  between  P6  and  Pf  increases,  and  is  inversely 
as  the  square  of  the  number  of  strokes. 

A  glance  at  the  cross-hatched  portion  of  Fig.  10  shows 
that  the  pressure  upon  the  crank-pin  is  by  no  means  uni- 
form, being  greatest  at  the  point  of  cut-off. 

We  further  know  that  the  angular  position  of  the  con- 
necting-rod causes  the  straight  line  A  B  to  become  a  curve. 
The  effect  of  the  connecting-rod  is  discussed  at  considerable 
length  in  The  Richards  Steam-engine  Indicator.  The  curve 
departs  more  widely  from  the  straight  line  as  n,  the  ratio 


OF  THE  STEAM-ENGINE.  83 

of  the  connecting-rod  to  the  crank,  becomes  less.  We  see 
then  that,  for  early  cutting  off  of  the  steam,  either  the  recip- 
rocating parts  should  be  made  heavy  or  the  number  of  revo- 
lutions increased  or  decreased  until  the  assumed  weight  G 
of  the  reciprocating  parts  is  obtained. 
Transposing  equation  (98),  we  have 


b      /,  (99) 

GT 

which  enables  us  to  determine  approximately  the  number 
of  strokes  per  minute  for  any  assumed  pressure  and  weight 
of  reciprocating  parts,  which  will  give  a  nearly  uniform 
stress  on  the  crank-pin. 

Pf  may  be  determined  from  Pb  with  sufficient  approxima- 
tion for  practical  purposes  from  Mariotte's  or  Boyle's  law 
for  gases  that  the  pressures  are  inversely  as  the  volumes. 

The  initial  and  final  pressures  can  be  taken  from  an  indi- 
cator diagram,  and  should  be  determined  after  the  erection 
of  the  engine. 

Example.  —  To  determine  the  proper  weight  of  the  recipro- 
cating parts  of  a  horizontal  engine,  data  as  follows  (non- 
condensing  cut-off  =  £  stroke)  : 

Pb  =  125  pounds  per  square  inch. 
Pf  =  25  pounds  per  square  inch. 
d  =  12  inches. 
JV=  200  per  minute. 


Substituting  in  formula  (98),  we  have  the  weight  =- 
4612.3x100x144 


40000 


=  3321  pounds. 


Noting  that  this  weight  is  very  large,  and  assuming  800 
pounds  as  the  approximate  weight  of  the  'reciprocating  parts 


84  THE  RELATIVE  PROPORTIONS 

of  a  steam-engine  of  one  foot  stroke  and  one  foot  diameter 
of  cylinder,  we  have,  substituting  in  formula  (99)  the 
proper  number  of  strokes, 

N=  67.914  x  12  J—      -  =  407  strokes  per  minute. 
^  800  x  -g- 

In  an  unbalanced  vertical  engine  the  weight  of  the  recip- 
rocating parts  lessens  the  resistance  with  which  the  piston 
leaves  the  upper  end  of  the  cylinder  and  adds  to  the  resist- 
ance with  which  it  leaves  the  lower  end  of  the  cylinder, 
thus  shifting  the  line  A  B,  Fig.  10,  parallel  to  itself  above 
or  below,  but  not  introducing  any  greater  irregularity  of 
pressure  upon  the  crank-pin  during  one  stroke. 


OF  THE  STEAM-ENGINE.  85 


CHAPTER   VIII. 

(36.)  The  Single  Crank. — The  crank  is  made  of  either 
cast  or  wrought  iron  or  steel ;  the  first-named  metal  is  but 
little  used,  and  should  be  avoided  in  any  but  the  very  rough- 
est machinery,  because  it  is  very  liable  to  hidden  defects,  is 
much  weaker  and  has  a  lesser  modulus  of  elasticity,  thus  re- 
quiring to  be  heavier  than  wrought  iron  or  steel.  Further, 
it  will  not  admit  of  the  crank-pin  being  "  shrunk  in  "  to  the 
eye  without  great  danger  of  cracking. 

This  latter  fact  is  illustrated  very  practically  by  an  oc- 
currence described  by  William  Pole  in  his  Lectures  on  Iron 
as  a  Material  of  Construction,  p.  123.  The  italics  are  ours : 

"You  have  probably  heard  of  the  process  of  drawing 
lead  tube  by  forcing  it  in  a  semi-fluid  (or  sometimes  in  a 
nearly  solid)  state  through  a  small  annular  hole.  The  lead 
is  contained  in  a  cylinder  and  pressed  upon  by  a  piston,  and 
the  force  required  is  enormous,  amounting  to  50  or  60  tons 
per  square  inch. 

"  The  practical  difficulty  of  getting  any  cylinder  to  with- 
stand the  pressure  was  almost  insurmountable.  Cast  iron 
cylinders  12  inches  thick  were  quite  useless ;  they  began  to 
open  in  the  inside,  the  fracture  gradually  extending  to  'the  out- 
side, and  increased  thickness  gave  no  increase  of  strength. 

"Cylinder  after  cylinder  thus  failed,  and  the  makers 
(Messrs.  Eaton  &  Amos)  at  length  constructed  a  cylinder 
of  wrought  iron  8  inches  thick ;  after  using  this  cylinder 
the  first  time,  the  internal  diameter  was  so  much  increased 
by  the  pressure  that  the  piston  no  longer  fitted  with  a  suffi- 
cient closeness.  A  new  piston  was  made  to  suit  the  enlarged 
cylinder ;  and  a  further  enlargement  occurring  again  and 
again  with  renewed  use,  the  constant  requirement  of  new 


86  THE  RELATIVE  PROPORTIONS 

pistons  became  almost  as  formidable  an  obstacle  as  the 
failure  of  the  cast-iron  cylinder. 

"  The  wrought-iron  cylinder  was  on  the  point  of  being 
abandoned,  when  Mr.  Amos,  having  carefully  gauged  the 
cylinder  both  inside  and  out,  found  to  his  surprise  that 
although  the  internal  diameter  had  increased  considerably  the 
exterior  retained  precisely  its  original  dimensions.  He  con- 
sequently persevered  in  the  construction  of  new  pistons,  and 
found  ultimately  that  the  cylinders  enlarged  no  more,  and 
so  the  last  piston  continued  in  use  for  many  years. 

"  Here,  therefore,  the  permanent  set  operated  first  in  the 
internal  portions  of  the  metal ;  as  they  expanded  it  was  then 
gradually  extended  to  the  surrounding  layers,  and  so  at  last 
sufficient  material  was  brought  into  play  with  perfect  elas- 
ticity, not  only  to  withstand  the  strain,  but  to  return  back 
to  the  normal  state  every  time  after  its  application,  and  thus, 
by  the  spontaneous  and  unexpected  operation  of  what  was 
then  an  unknown  principle,  an  obstacle  apparently  insur- 
mountable, and  which  threatened  at  one  time  to  render 
much  valuable  machinery  useless,  was  entirely  overcome." 

It  is  this  very  property  of  wrought  iron  or  soft  steel 
which  renders  it  useful  for  cranks ;  the  metal  when  heated 
possesses  increased  ductility  or  viscidity,  and  adapts  itself  to 
the  strain  put  upon  it,  while  shrinking  in  a  very  perfect 
manner. 

The  following  expansions  in  length  for  one  degree  Fah- 
renheit are  given  in  Table  XIV.,  p.  15,  of  Box's  Practical 
Treatise  on  Heat: 

Cast  iron 000006167 

.     Steel 000006441 

Wrought  iron 000006689 

Neglecting  cast  iron  as  being  unsuitable  for  cranks,  let  ua 
take  up  the  case  of  a  wrought-iron  crank. 


OF  THE  STEAM-ENGINE.  87 

The  value  of  E  for  wrought  iron — i.  e.,  the  hypothetical 
weight  which  would  stretch  a  bar  one  square  inch  in  area 
to  twice  its  original  length  if  it  were  perfectly  elastic — is 
28000000  pounds. 

If  we  take  the  rupturing  strain  in  tension  for  wrought 
iron  at  ordinary  temperatures  at  50000  pounds  per  square 
inch  area, 

Letting  x  =  the  extension  at  the  point  of  rupture,  we  have 
1  :  x  : :  28000000  :  50000,  and 

x  = =  .0017857  of  its  length. 

2800 

If  further  we  divide  this  amount  by  the  lengthening  of  a 
bar  for  one  degree  Fahr.,  we  have 

.0017857    =  267  Agrees  Fahr., 


.000006689 

and  we  see  that  if  it  were  not  for  the  viscosity  of  wrought 
iron,  a  bar  heated  267°  and  fastened  so  as  not  to  be  able 
to  contract  would  rupture  in  cooling,  and  that  further  a 
difference  of  26.7  degrees  Fahr.  would  in  the  contraction 
resulting  strain  a  bar  5000  pounds  per  square  inch  area. 

Example. — Let  us  assume  the  diameter  of  that  part  of  the 
crank-pin  which  is  to  be  inserted  into  the  eye  of  the  crank  at 
5  inches,  and  further  that  the  eye  of  the  crank  is*  bored  to 
a  diameter  of  4.98  inches — an  accuracy  which  it  is  quite 
practicable  to  obtain  in  any  good  machine-shop. 

We  then  have  for  the  required  number  of  degrees  through 
which  the  eye  of  the  crank  must  be  raised 

09 

630  degrees  Fahr. 

.000006689x5 

More  accurately,  4.98  should  have  been  used  in  the  place 
of  5  inches. 


88  THE  RELATIVE  PROPORTIONS 

Knowing  that  at  a  temperature  of  977  degrees  Fahr.  iron 
is  just  visibly  red,  we  may  use  the  following  formula  to 
determine  the  difference  in  the  diameters  of  the  eye  of  the 
crank  and  of  the  inserted  part  of  the  crank-pin : 

Let  dc  =  diameter  of  inserted  part  of  pin, 
"  de  =  required  diameter  of  eye ; 

we  have,  with  sufficient  approximation, 
d.-d. 


.000006689^ 


900, 


Should  this  formula  give  a  greater  difference  than  is 
necessary  in  practice,  reduce  it  according  to  judgment  by 
using  less  heat,  unless  a  particularly  strong  grip  of  the 
eye  upon  the  pin  is  desired.  A  high  heat  is  apt  to  warp  a 
forging,  and  no  more  should  be  used  than  is  necessary.  This 
formula  cannot  be  regarded  as  accurate,  and  should  be  used 
as  a  guide  only ;  it  applies  with  sufficient  approximation  to 
machinery  steel. 

If  the  entering  part  of  the  crank-pin  be  made  very  slightly 
taper  and  larger  than  the  eye  of  the  crank,  and  the  pin  be 
forced  in  by  strong  hydraulic  pressure,  we  avoid  the  risk  of 
warping  occasioned  by  heat,  and  secure  almost  as  good  a 
result  so  far  as  strength  of  grip  is  concerned. 

Crank-pins  are  sometimes  fastened  by  a  key  "  cutter  "  at 
the  back,  and  sometimes  held  by  a  nut. 

Special  care  must  be  taken  to  have  a  sufficient  thickness 
of  metal  around  the  eye  to  hold  the  pin  firmly.  In  prac- 
tice the  thickness  of  the  ring  around  the  eye  is  usually 
taken  equal  to  one-half  the  diameter  of  the  eye.  The  depth 
of  the  eye  is  usually  from  1  to  2  times  its  diameter,  thus 
giving  a  cross-section  of  ring  around  the  eye  of  from 


OF  THE  STEAM-ENGINE.  89 

to  2T5^g-  greater  area  than  that  of  the  inserted  part  of  the 
crank-pin  at  right  angles  to  its  axis. 

In  some  cases  the  pin,  crank  and  crank-shaft  are  forged 
in  one  solid  piece  out  of  wrought  iron  or  made  of  cast  steel 
in  one  solid  mass ;  this  method  is  more  expensive,  but  more 
satisfactory. 

The  web  of  the  crank  is  that  part  connecting  its  hub  and 
eye,  and  is  made  of  many  differing  shapes,  and  always  with 
its  cross-section  increasing  from  the  eye  to  the  hub.  Econ- 
omy of  material,  as  will  presently  be  shown,  dictates  that 
this  cross-section  be  increased  by  an  increase  of  the  breadth 
of  the  face  of  the  crank — i.  e.,  in  a  direction  at  right  angles 
to  the  centre  line  of  the  crank-pin,  rather  than  in  the  direc- 
tion of  the  crank-pin. 

If  now  we  assume  that  the  crank  has  a  constant  thick- 
ness in  the  direction  of  the  centre  line  of  the  crank-pin, 
the  longitudinal  profile  of  the  web  of  the  crank  should  be 
a  parabola  (Weisbach's  Mechanics  of  Engineering,  sec.  iv., 
art.  251-253),  with  its  vertex  at  the  centre  of  the  pin, 
and,  providing  we  neglect  the  torsional  strain  on  the  web 
due  to  the  moment  of  the  crank-pin,  will  be  of  uniform 
strength. 

The  torsion  of  the  web  due  to  the  force  acting  at  the 
crank-pin  is  much  slighter,  in  fact,  than  might  at  first  appear, 
as  the  pin  is  fastened  in  the  connecting-rod  as  well  as  the 
crank ;  and  if  the  pin  is  a  little  loose,  as  it  never  should  be, 
its  springing  tends,  by  throwing  the  centre  of  pressure  nearer 
the  crank,  to  reduce  the  moment.  We  can,  therefore,  neg- 
lect the  torsion  of  the  web,  and  consider  it  as  a  beam  re- 
quired to  be  of  uniform  strength,  fixed  at  one  end  and  loaded 
at  the  other. 

To  trace  the  parabolic  outline  of  the  crank  would  be 
tedious  and  result  in  an  ugly  shape,  but  the  property  of  the 
tangent  of  a  parabola  permits  us,  with  an  increase  of  less 


90  THE  EELATIVE  PROPORTIONS 

than  six  per  cent,  of  material,  to  locate  correctly  the  straight 
sides  of  the  web. 

The  interior  diameter  of  the  hub  of  the  crank  depends 
upon  the  diameter  of  the  main  or  crank-shaft ;  the  depth  of 
the  hub  is  usually  made  equal  to  the  diameter  of  the  shaft, 
or  somewhat  greater,  and  the  thickness  of  metal  around  the 
shaft  equal  to  J  of  its  diameter. 

These  proportions  are  varied  very  frequently,  and  are  to 
a  great  extent  a  matter  of  judgment  with  the  designer. 

The  crank  is  frequently  shrunk  on  the  shaft,  just  as  de- 
scribed in  the  case  of  the  crank-pin,  and  in  such  a  case 
similar  precautions  must  be  taken. 

(37.)  Wrought-Iron  Single  Crank.— Referring  to 
Fig.  5,  we  see  that  in  addition  to  a  constant  stress  S  in  a 

FIG.  5. 


horizontal  direction  the  crank  is  subjected  to  an  alternate 

o 

maximum  stress  in  tension  and  compression,  Si" 


which,  for  the  values  of  n  =  4  to  8,  gives  Si  =  0.26  to  0.13  S. 
See  formula  (38),  Art.  19. 

Let  S  =  the  stress  in  pounds  upon  the  crank  at  extremity, 

tending  to  cause  flexure. 
"   Si  =  the  stress  in  pounds  upon  crank,  tending  to  cause 

extension  or  compression. 
"   -F=the  cross-section  of  the  crank  in  inches. 
"    b  =  the  assumed  thickness  of  the  crank  in  inches  at 

right  angles  to  its  face. 


OF  THE  STEAM-ENGINE.  91 

Let    v  =  the  variable  width  of  the  face  of  the  crank  in  inches. 
"     x  =  the  variable  length  of  the  crank  from  the  centre 

of  the  eye. 
"    JF=  the  measure  of  the  moment  of  flexure  of  the  cross- 

section  of  the  crank  at  any  point. 
"     2"=  the  safe  strain  per  square  inch  upon  the  crank  - 

5000  pounds. 

"      e  =  the  half  width  of  the  face  of  the  crank  =  —  . 

"      r  =  the  radius  of  the  crank  in  inches  =  —  . 

We  have,  if  we  suppose,  as  is  usually  assumed,  the  con- 
necting-rod to  be  of  infinite  length  (i.  e.,  n  =  oo  ),  Si  =  0,  but 
/%  is  too  large  to  be  neglected  in  practice. 

Placing  ._"       r->  +  lP  .     (102) 

See  Weisbach's  Mechanics  of  Engineering,  sec.  iv.,  art.  272. 
Therefore,  J=f  +|    f  •  (103) 

7p  /» 

For  a  rectangular  cross-section  —  =  —  . 

W     v 

Therefore,  F=-(  -  -  -  +-V         (104) 

irvyirri    +) 

Substituting  for  F  its  value  bv,  we  have 

,       81       I          6x\ 
bv  =  —4  --  +  —  I. 
T\V^=l      v) 

Therefore,  *-  -  ?  -  +J         ^          +ggg.  (105) 
^1     >462T2(n2-l)     bT 


We  have  then  the  means  of  computing  the  value  of  V  for 
a  series  of  assigned  values  of  b  and  x. 


92 


THE  RELATIVE  PROPORTIONS 


We  see  further  that  the  first  two  terms  of  the  second 
member  of  equation  (105)  will  give  small  values  which  do 

not  greatly  affect  the  value 


FIG.  11. 


of  v. 


sume 


If  in  (105)  we  as- 

L     r  ^ 

x  =  —  =  -,  we   have 

the  width  of  the  face  of  the 
crank  at  a  point  midway 
between  the  centres  of  the 
eye  and  hub,  and  equation 
(105)  becomes 

8 


SSL 

(] 


If  we  suppose  n  =  oo  ,  equa- 
tion (106)  becomes 


[ML 


(107) 


In  making  use  of  equa- 
tions (105)  and  (106)  it  is 
most  convenient  to  calcu- 
late the  value 


separately.  If  in  (107)  we 
substitute  the  values  of  S 
and  T%  we  have 


0.0153d 


(108) 


OF  THE  STEAM-ENGINE.  93 

and  for  a  uniform  pressure  P  we  have,  in  terms  of  the 
horse-power, 


Vl  =  10.864  .  (109) 

*    bN 

If  now  we  assume  the  value  b  =  the  thickness  of  the  web 
as  uniform  throughout  its  length,  and  recollecting  that  the 
tangent  to  a  parabola  intersects  the  axis  of  abscissa  at  a  dis- 
tance from  the  origin  (vertex)  equal  to  the  abscissa  of  the 
point  of  tangency,  we  have  a  ready  means  of  graphically 
constructing  the  web  of  the  crank,  Fig.  11. 

From  equation  (106)  or  (108)  (the  first  is  the  more  ex- 
act) calculate  the  value  of  vlt  Draw  the  centre  line 
ABC  through  the  centres  of  the  eye  and  hub.  Bisect 
D  C  in  B  and  lay  off  D  A  =  D  B  ;  at  the  point  B  erect 
the  double  ordinate  vly  and  through  the  extremities  of  this 
ordinate  and  through  the  point  A  draw  the  lines  A  F  G  and 
A  H  K  to  intersections  with  eye  and  hub. 


CHAPTER    IX. 

(88.)  Steel  Single  Crank.— Formula?  (105)  and  (106) 
apply  in  the  same  manner,  with  the  exception  that  T=  9000 
pounds  per  square  inch  for  steel.  Formula  (107)  becomes 
for  this  value  of  T,  and  the  substitution  of  the  value  of  S, 


=  0.0114^ 


and  for  an  uniform  pressure  P  we  have,  in  terms  of  the 
horse-power, 


(Ill) 


94  THE  RELATIVE  PROPORTIONS 

Examples.  —  Let         .L  =  48  inches. 
d  =  32  inches. 
"         Pb  =  P=  40  pounds  per  square  inch. 


"   (HP)  =  156  approximately. 
"          JV=  40  per  minute. 
"  6  =  7  inches. 

We  have  #  =  0.7854x1024x40  =  32170  pounds. 

For  a  wrought-iron  crank,   ^=5000  pounds  per  square 
inch. 

Substituting  in  formula  (1 06),  we  have 

_  32170 

Vl~  2x7x5000x4.899 


(32170)2  3x32170x48 

4  x  49  x  25000000  x  24      2  x  7  x  5000 


=  0.094  + 1/0.0088  +  66.18  =  0.094 +  8.25  =  8.344  inches. 
If  we  substitute  in  formula  (109),  we  have 

Vi  =  10.864X/-^L  =  8.H  inches. 

\  7x40 

We  see  that  the  difference  between  the  results  of  formulae 
(106)  and  (109)  is  trifling,  and  this  difference  will  be  less  for 
all  values  of  n  >  5  ;  it  will  be  greater  for  all  values  of  n<  5. 

For  a  steel  crank  we  should  take  b  =  6"  and  T=9000 
pounds  per  square  inch. 

Substituting  in  formula  (111),  we  have 


The  use  of  these  values  of  Vi  in  the  graphical  construc- 
tion of  the  web  of  the  crank,  Fig.  11,  Art.  37,  is  obvious. 


OF  THE  STEAM-ENGINE. 


95 


FIG.  12. 


FIG.IS. 


The  use  of  formula  (105)  for  the  computation  of  a  series 
of  values  of  v  for  assigned  values  of  x  and  b  is  tedious,  but 
easy  to  understand. 

(39.)  Cast-Iron  Cranks. — Although  subject  to  a  greater 
liability  to  accidental  frac- 
ture than  wrought  iron  or 
steel,  cast-iron  cranks  are 
frequently  used  in  the 
cheaper  forms  of  engines. 

Fig.  12  gives  a  rather 
neat-looking  unbalanced 
crank,  and  Fig.  13  is  an 
example  of  the  disc  crank, 

which  allows  of  being  balanced  with  great  ease  by  means 
of  a  counter-weight  placed 
on  the  opposite  side  from 
the  pin. 

The  proportions  of  cast- 
iron  cranks  depend  to  a 
great  extent  upon  the  cha- 
racter of  the  iron  used. 
Formulae  .  can  hardly  be 
applied  to  so  uncertain  a 
material  as  cast  iron. 

(40.)  Keys  for  Shafts.— In  every  case  a  key  is  used 
to  prevent  the  crank  from  slipping  on  the  shaft.  The  best 
position  for  a  key  is  in  line  with  and  between  the  centres 
of  the  eye  and  hub  of  the  crank.  If  more  than  one  key  is 
used,  their  combined  shearing  strength  should  be  equal  to 
that  of  a  single  key.  Great  care  should  be  taken  that  the 
sides  of  the  key  fit  the  keyway  perfectly,  and  that  these 
sides  be  perfectly  parallel,  not  taper.  The  very  slight  taper 
given  to  the  key  should  be  at  the  top. 

Considering  the  case  of  a  single  key, 


96 


THE  RELATIVE  PROPORTIONS 


Let  bi  —  its  breadth  in  inches. 

"      I  =  its  length  in  inches. 

"      t  =  its  thickness  in  inches. 

"    K=  the  safe  shearing  strain  per  square  inch  area. 

"     d  =  the  diameter  of  the  steam -cylinder  in  inches. 

"     L  =  the  length  of  stroke  in  inches. 

"   Pb  =  the  steam-pressure  per  square  inch. 

"     di  =  the  diameter  of  the  shaft  in  inches. 
Since  the  moment  of  the  safe  shearing  strain  of  the  key 
should  be  equal  to  the  maximum  moment  upon  the  crank, 

we  have 

_._  0.7854 

2 


Therefore, 


(112) 


Or,  if  we  suppose  the  steam-pressure  to  be  uniform,  we 
have,  in  terms  of  the  horse-power  (HP),  and  number  of 
strokes  per  minute  Nt 


396000 


(HP} 


FIG.  14. 


(113) 


To  determine  the  proper  thickness  of  the  key  t  we  may 

adopt  the  following  method : 

Let  the  key  be  sunk  one- 
half  of  its  thickness  in  the 
shaft,  Fig.  14,  and  let  the 
thickness  of  the  key  be  such 
that  the  shearing  surface  of 
the  shaft  on  a  line  with  the 
inside  surface  of  the  key 
N  M  equals  the  shearing  sur- 
face of  the  key  ON.  If  t 
be  not  determined  graphi- 
cally, it  can  be  determined 

approximately  in  an  analytical  way. 


OF  THE  STEAM-ENGINE.  97 

We  have  t  =  2KL  =  2(  CK-  CL).  (114) 


If  now  we  substitute  these  values  of   CK  and  CL  in 
formula  (114),  we  have 


and  reducing  we  have,  with  sufficient  approximation, 

4&L2    105^ 

£  =  -7-  +-77-+  etc.  (115) 

di       df 

Example.  —  Let  e?4  =  16  inches. 
"   ^  =  3  inches. 

Substituting  in  formula  (115),  we  have 

4x(3)2    10x(3)4  810 

t  =  —  —  +  -  ~~—  8J5*  --  =  2.45  niches. 
16         (16)s  4096 

Where  more  than  one  key  is  used  the  sum  of  their 
widths  should  equal  the  width  of  a  single  key  fitted  to 
withstand  the  given  stress. 

The  thickness  of  the  keys  will  be  much  less,  as,  for  in- 
stance, two  keys  1^-  inches  broad  and  -^  of  an  inch  thick 
will  take  the  place  of  the  single  key  given  in  the  example 
with  equal  safety. 


98  THE  RELATIVE  PROPORTIONS 

It  is  a  point  worthy  of  particular  notice  that,  neglecting  the 
last  term  of  equation  (115},  we  see  that  the  thickness  of  keys  for 
shafts  varies  directly  as  the  square  of  their  breadth,  and  there- 
fore that  the  use  of  two  or  more  keys  is  attended  with  great 
economy  of  material  in  the  keys,  and  less  reduction  of  the 
cross-section  of  the  shaft,  while  the  safety  is  equally  as  great. 

In  establishing  the  proper  thickness  of  metal  for  the  hub 
of  the  crank  the  reduction  due  to  the  keyways  must  be 
considered. 

(41).  Wrought-Iron  Keys  for  Shafts.—  If  in  formula 
(112)  we  place  if  =5000  pounds  per  square  inch,  we  have 

fti-  0.000157^-^.  (116) 

Idt 

Letting  K=  5000  pounds  per  square  inch,  we  have,  from 
formula  (113), 

^  =  79.21  (117) 


Example.  —  Let         d  =  32  inches. 

"         Pb  =  P=  40  pounds  per  square  inch. 

L-  48  inches. 
"  1  =  20  inches. 

"          d<  =  16  inches. 

J\T=40  per  minute. 
"   (HP)  -  156  approximately. 

Substituting  in  formula  (116),  we  have 

6l  i  0.000157  1024x4°x48  -  0.96  inches. 
16x20 

Substituting  in  formula  (117),  we  have 


OF  THE  STEAM-ENGINE.  99 

We  have  also  for  the  thickness  of  the  key,  from  formula 
(115), 

t  =  4x0.92 

16 

(42.)  Steel  Keys  for  Shafts.—  Since  the  shearing 
strength  of  machinery  steel  is  found  to  be  f  of  its  tensile 
strength,  we  have  for  a  safe  shearing  stress  f  of  9000 
pounds  =  6750  pounds  per  square  inch.  Substituting  this 
value  for  JTin  formulae  (113)  and  (114),  we  have 


(118) 
i 

or 

^  =  58.667^1  (119) 


Comparing  formulae  (117)  and  (119),  we  find 

bv  for  steel  _    =  58.667  ^Qy4 
£>!  for  wrought  iron       79.2 

Example.  —  Data  as  before  in  Art.  (41).  We  can  calcu- 
late as  there  shown,  or  abbreviate  by  taking  74  per  cent,  of 
the  breadth  of  a  wrought-iron  key  for  the  breadth  of  a  steel 
key  of  equal  strength, 

h  =  0.74  x  0.96  =  0.71  inches. 

The  thickness  would  be  the  same  as  before  for  a  wrought- 
iron  shaft,  and  for  a  steel  shaft,  formula  (115),  we  have 


16 

This  latter  value  is  too  small  for  convenience,  and  the 
key  should  be  made  of  J  inch  thickness,  or  as  would  be 
advisable  greater  in  both  cases. 

(43.)  The  Crank  or  Main  Shaft.—  The  first  requisite 
of  the  crank-shaft  is  rigidity.  Any  flexure  will  be  accom- 


100  THE  RELATIVE  PROPORTIONS 

panied  by  injury  and  final  destruction  to  the  bearings  of  the 
crank-pin  and  of  the  shaft  itself. 

The  stresses  to  which  the  crank-shaft  is  subjected  are  as 
follows : 

(1.)  At  the  beginning  of  each  stroke  it  receives  the  thrust 
due  to  the  pressure  of  steam  upon  the  piston-head,  which 
in  some  cases  is  a  veritable  blow,  as  will  be  noticed  when 
the  journals  are  not  properly  fitted,  and  also  with  some 
valve-motions  without  compression. 

(2)  At  the  middle  of  the  stroke  the  crank-shaft  is  sub- 
jected to  a  maximum  torsional  stress  (whose  moment  is 
measured  by  the  length  of  crank  multiplied  by  the  steam- 
pressure  on  the  whole  of  the  piston),  in  addition  to  the  thrust 
due  to  the  whole  steam-pressure  on  the  piston. 

(3.)  When  a  fly-wheel  propellor  or  paddle-wheels  are 
attached  to  the  shaft  it  is  deflected  by  their  weight,  as  well 
as  strained  by  the  torsional  stress  mentioned  above. 

These  three  cases  will  be  discussed  separately. 

(M.)  Shaft  Subjected  to  Flexure  only.— Round 
shafts  only  are  used  for  steam-engine  crank-shafts,  and  will 
therefore  be  the  only  form  of  shaft  considered. 

Shafts  are  subject  to  stress  in  flexure  only,  either  when  at 
rest  or  in  some  cases  at  the  dead  points — that  is,  when  no 
fly-wheel  is  used.  The  parts  subject  to  flexure  are  the  over- 
hanging ends  of  the  shaft  to  which  the  crank  propellor  or 
paddle-wheel  is  attached,  or  between  those  bearings  which 
support  the  fly-wheel  propellor  or  paddle-wheel. 

Let    T=the  safe  stress  per  square  inch  upon  the  exterior 

fibres  of  the  shaft. 
"    JF=the  measure  of  the  moment  of  flexure  for  a 

round-shaft  =  — -. 
64 

*•    d4/  =  the  required  diameter  of  the  shaft  to  resist  flexure, 
in  inches. 


OF  THE  ,ST^AM-MrGIN^      >V     /!;  101 

Let    I  =  the  length  of  that  part  of  the  shaft  under  con- 
sideration in  inches. 
"     G  =  the  load  upon  the  shaft  in  pounds. 

We  have  (Weisbach's  Mechanics  of  Engineering,  sec.  iv., 
art.  235),  for  a  shaft  fixed  at  one  end  and  loaded  at  the 
other, 


Transposing, 


32        I 
32     Ol 

=   7T  T' 


(120) 


(121) 


FIG.  15. 


(45.)  Wrought-Iron   Shaft  Flexure   only.— In  the 

case  of  a  wrought-iron  shaft  27=5000  pounds  per  square 
inch,  and  formula  (120)  becomes 


102  'j?&E  RELATIVE  PROPORTIONS 

<V  =  -  002037  <&  (122) 

Formula  (121)  becomes 

d4/=  0.1268  i/~GL~  (123) 


In  the  case  of  an  overhanging  weight,  as  at  E,  Fig.  15, 
these  two  formulae  can  be  applied  directly  by  substituting  E 
for  G  and  13  for  I. 

In  the  case  of  a  load  C  between  two  supports  B  and  Z), 
we  can  determine  the  reactions  at  B  and  D  as  follows  : 

Let  /i  =  the  distance  BC  in  inches. 
"    /2  =  the  distance  CD  in  inches. 
"    C=  the  weight  in  pounds. 
"  B  —  the  reaction  at  support  B  in  pounds. 
"  D  =  the  reaction  at  support  D  in  pounds. 


We  have  Cl 

Therefore,  D  =  --.  (124) 

(  li  +  4) 

We  have  then  to  substitute  the  derived  value  of  D  for  G 
and  4  for  /,  in  equation  (122)  or  (123),  in  order  to  determine 
the  diameter  for  flexure  of  the  part  CD  of  the  shaft.  Simi- 
larly, we  can  treat  the  part  BC  of  the  shaft. 

The  overhanging  part  AB  of  the  shaft  should  be  estimated 
from  the  centre  of  the  crank-pin,  and  for  G  the  maximum 
pressure  of  the  steam  upon  the  piston-head  =  0.7854P6cf 
should  be  substituted. 

Making  these  substitutions  in  formula  (122),  we  have 

(125) 
(126) 


=   .6. 
If  we  suppose  the  pressure  Pb  uniform,  we  have 

cV-806J^2*. 
X/iV 


OF  THE  STEAM-ENGINE.  103 


CHAPTEE    X. 

(46.)  Steel  Shaft  Flexure  only. — For  a  steel  shaft 
T=  9000  pounds  per  square  inch,  and  formula  (120)  becomes 

d4/  =  0.00113  Gl  (127) 

Formula  (121)  becomes 

d«  =  0.1042  i/~GT  (128) 

The  method  of  applying  these  two  formulae  is  the  same 
as  in  Art.  (45),  equation  (124),  etc. 

For  the  overhanging  part  of  the  shaft  AB,  Fig.  15,  to 
which  a  single  crank  is  attached,  we  have  by  substitution  in 

formula  (127) 

dj  -  0.000889  PbcP  1.  (1 29) 

If  in  this  we  suppose  the  steam-pressure  Pb  to  be  uniform, 
we  have 


dj  =  448.18^^- 

(47.)  Shaft  Subjected  to  Torsion  only.— Shafts  are 
subjected  to  a  stress  in  torsion  from  the  point  where  the 
power  is  received  to  the  point  at  which  it  is  given  off,  and 
not  beyond  that  point. 

The  torsional  stress  in  crank-shafts  is  zero  at  the  dead 
points,  and  a  maximum  when  the  crank  stands  at  right 
angles  to  the  centre  line  of  the  steam-cylinder  for  single 
cranks.  (For  double  and  treble  cranks  see  Arts.  54  and  55.) 

We  need  only  to  consider  the  maximum  torsional  stress 
upon  the  shaft,  leaving  out  of  consideration  the  deflecting 
stress  occurring  at  the  point  of  maximum  torsional  stress, 
and  due  to  the  angular  position  of  the  connecting-rod. 
(See  Art.  37.) 


104  THE  RELATIVE  PROPORTIONS 

Let  r   =  the  length  of  the  crank  in  inches  =  — . 

2 
"   ^«  =  the  required  diameter  of  the  shaft  to  resist  torsion, 

in  inches. 
"    T  =  the  safe  stress  per  square  inch  upon  the  exterior 

fibres  of  a  round  shaft. 
"   S  =  0.7854P6d2  =  the  stress    at  the  extremity   of   the 

crank — i.  e.,  upon  the  crank-pin. 
We  have  (Weisbach's  Mechanics  of  Engineering,  sec.  iv., 

art.  264) 

3==16/Sr 


Therefore,  substituting  for  S  and  r  the  values  given  above, 
^8  =  2;ZVfZ, 

/TT       '  ' 

and  <4  =  1.26^T  (m) 

If  in  formula  (131)  we  consider  the  pressure  of  the  steam 
to  be  uniform,  we  have,  in  terms  of  the  horse-power," 

e^3  =  1008660.^-2  (133) 

(48.)  Wrought-Iron  Shaft  Torsion  only. — For  wrought 
iron  we  have,  with  a  factor  of  safety  of  10,  T*=  5000  pounds 
per  square  inch,  and  formula  (131)  becomes 

d«3  =  0.0004P6dlL,  (134) 

and  formula  (132)  becomes 

Formula  (133)  becomes 

(  JTJ-*\ 

7  3       OfM   TO^"^       '  f"\  Q£J\ 

Ujt    =ZU1.«U .  llOD) 

N 

(49.)  Steel  Shaft  Torsion  only. — Since  the  stress  upon 
a  steel  shaft  at  the  outer  layer  of  fibres  is  of  the  nature  of 


OF  THE  STEAM-ENGINE.  105 

a  shearing  stress,  we  can  take,  with  a  factor  of  safety  of  10, 
T=  6750  pounds  per  square  inch. 
Formula  (131)  becomes 

dj  -  0.0002964P»«M.  (137) 

Formula  (132)  becomes 

d«  -  0.06667y/P^Z:  (138) 

Formula  (133)  becomes 

eV-  149.43^2.  (139 

N 

(50.)  Shaft  Submitted  to  Combined  Torsion  and 
Flexure. — The  most  frequent  case  for  crank-shafts  is  where 
they  are  submitted  at  the  same  time  to  stresses  of  flexure 
and  torsion. 

Let  T=the  safe  stress  per  square  inch. 
"    $  =  the  whole  stress  of  the  pressure  of  the  steam  upon 

the  piston-head,  in  pounds. 
"    G  —  the  load  causing  flexure,  in  pounds. 
"     /  =  the  distance  in  inches  of  the  point  of  application 

of  the  load  to  the  point  of  support. 

"    r  =  the  length  of  the  crank  in  inches  —  — . 

2i 

"    d±  =  the  required  diameter  of  the  shaft,  in  inches,  to  with- 
stand the  combined  stresses  of  flexure  and  torsion. 
We  have  ( Weisbach's  Mechanics  of  Engineering,  sec.  iv., 
art.  277)  the  following  approximate  formulae: 


(140) 


106  THE  RELATIVE  PROPORTIONS 

•mi 


or 

dt  —  (141) 


Considering  formulse  (140)  and  (141),  we  observe  that 

Q9  C1? 
c^.  Kefer  to  Art.  (47)  ;  and  that  —   Tf-  =  rfv8.    Ke- 

Tt  TTjT 

fer  to  Art.  (44). 

Substituting  these  values,  we  have 


*-r  ,'  -,.          (143> 

M4 


As  in  both  formulae  (142)  and  (143)  c?4  is  greater  than 
either  d4/  or  d4t,  we  must,  in  order  to  obtain  the  first  approx- 
imation, substitute  for  d4  in  the  second  member  the  greater 
resulting  diameter  from  a  consideration  of  the  stresses  in 
flexure  and  torsion  singly. 

A  single  approximation  will  generally  be  sufficient  for 
practical  purposes. 

Therefore  we  have  the  following  simple  rules  for  calcu- 
lating the  diameter  of  a  shaft  submitted  simultaneously  to 
torsion  and  flexure: 

FIRST.  Calculate  the  diameters  for  torsion  and  flexure 
singly. 


OF  THE  STEAM-ENGINE.  1Q7 

SECOND.  If  the  diameter  due  to  torsion  be  the  greater, 
divide  it  by  the  sixth  root  of  the  expression,  Unity  minus  the 
quotient  of  the  cube  of  the  diameter  due  to  flexure  divided  by 
the  cube  of  the  diameter  due  to  torsion,  and  the  result  will  be 
the  required  diameter. 

THIRD.  If  the  diameter  due  to  flexure  be  the  greater,  divide 
it  by  the  cube  root  of  the  expression,  Unity  minus  the  quotient 
of  the  sixth  power  of  the  diameter  due  to  torsion  divided  by 
the  sixth  power  of  the  diameter  due  to  flexure,  and  the  result 
will  be  the  required  diameter. 

Claudel,  Formules  a  Vusage  de  VIngenieur,  p.  288,  gives 
the  following  rule  :  "  Calculate  the  diameter  of  the  shaft 
to  resist  each  strain  separately.  Take  the  greatest  of  the 
two  values.  If  the  largest  diameter  is  given  by  the  effort 
of  torsion,  augment  it  by  £  to  -£$"  This  rule  gives  too 
small  values. 

Formulae  (142)  and  (143)  can  be  expanded  into  a  series 
giving  an  approximate  value.  Thus  : 


Therefore  (142)  becomes  when  torsion  gives  the  greater 
diameter 


and  in  the  same  manner  (143)  becomes  when  flexure  gives 
the  greater  diameter 


108  THE  RELATIVE  PROPORTIONS 

(51.)  Flexure  and  Twisting  of  Shafts.  —  The  shaft  is 
deflected  by  the  load  placed  upon  it,  and  also  twisted 
through  a  greater  or  less  angle  by  the  action  of  the  crank. 

Let  d±  =  the  diameter  of  the  shaft  in  inches. 
"    (r  =  the  load  in  pounds  at  the  extremity  of  the  part 

considered. 
"     I  =  the  length  of  that  part  of  the  shaft  under  consider- 

ation in  inches. 
"    E=the  modulus  of  elasticity  =  28000000  pounds  per 

square  inch  for  wrought  iron. 
"     a  •=  the  deflection  in  inches. 

We  have  (Weisbach's  Mechanics  of  Engineering,  sec.  iv., 
art.  217),  for  a  beam  fixed  at  one  end  and  loaded  at  the 
other, 

64       OT 


giving  the  deflection  from  a  straight  line. 

If  the  weight  of  the  shaft  be  taken  into  consideration,  we 
must  add  or  subtract  it,  as  the  case  may  require. 

For  the  angle  through  which  a  wrought-iron  shaft  will  be 
twisted  (Weisbach's  Mechanics  of  Engineering,  sec.  iv.,  art. 
263)  we  have,  letting  a°  =  the  number  of  degrees  and  the 
notation  be  as  before, 

(147) 
or  supposing  the  pressure  Pb  constant, 


a°  =  12.807  .  (148) 

J\Cl/4 

Example.  —  Fig.  15,  to  determine  the  proper  diameter  of 


OF  THE  STEAM-ENGINE. 


109 


a  wrought-iron  shaft  having  a  fly-wheel  C,  weighing  70000 
pounds,  supported  between  the  plumber-blocks  B  and  D. 


Fro.  15. 


Let        1  =  36  inches. 

"  /!  =  48  inches. 

"  fa  =  36  inches. 

"  Pb  =  P— 40  pounds  per  square  inch. 
"         d  =  32  inches. 

w  L  =  48  inches. 

"  N=  40  per  minute. 
"  (fiP)  - 156  approximately. 


The   Overhanging  Part  AB.  —  From  formula  (126)  we 
have  for  deflection 


=  806.7 


156x36 
48x40 


10 


110  THE  RELATIVE  PROPORTIONS 

From  formula  (136)  we  have 


Observing  that  the  diameter  due  to  flexure  is  the  greater, 
and  extracting  the  cube  root,  we  have 

dtf=  13.31  inches, 

and  using  formula  (143),  or,  more  conveniently,  (145),  we 
have 


giving  the  required  diameter  for  the  part  AB  of  the  shaft. 
The  Part  EC  of  the  Shaft.—  From  Art.  (45)  we  have  the 


reaction  at 

„      <%      70000x3     Qnn__ 
B  =  —  -  -  -  =  30000  pounds. 
Ja  +  4  7 

We  then  have,  formula  (122), 

dj  -  0.002037  x  30000  x  48  =  2933.28, 

dy=  14.31  inches. 
Using  formula  (145),  we  have 


giving  the  required  diameter  of  the  part  BC  of  the  shaft. 

The  Part  CD  of  the  Shaft.— By  substitution  in  formulae 
(124)  and  (123)  we  find  the  required  diameter  of  the  part 
CD  to  be  =  14.31  for  flexure  alone,  if  we  suppose  the  power 
to  be  taken  off  the  fly-wheel  and  neglect  the  stress  from  a 
belt  or  gearing. 

The  Part  DE.—lf  we  suppose  the  weight  at  .£=40000 


OF  THE  STEAM-ENGINE.  Ill 

pounds  and  13  =  36  inches,  we  have,  by  substitution  in  form- 
ula (123),  dt  =  14.31  inches  for  flexure  alone,  or,  by  substi- 
tution in  formula  (145),  if  we  suppose  the  shaft  submitted 
to  torsion  also, 

c?4  =  14.6  inches. 

For  torsion  only  d«  =  {/7S6.75  =  9.23  inches. 

(52.)  Comparison  of  Wrought-iron  and  Steel  Crank- 
Shafts. — Comparing  formulae  (128)  and  (123),  we  see 

For  steel  d4/ 0.1042 

For  wrought  iron  d4f  ~  0.1268  ~ 

Therefore,  a  steel  shaft,  to  withstand  the  same  stress  in 
flexure,  requires  to  be  but  0.82  of  the  diameter  and  0.67  of 
the  weight  of  a  wrought-iron  shaft. 

Comparing  formulae  (138)  and  (135),  we  have 

For  steel  d,t 0.06667 

For  wrought  iron  d^t      0.07368 

Therefore,  a  steel  shaft  withstanding  the  same  stress  in 
torsion  requires  to  be  0.90  the  diameter  and  0.81  the  weight 
of  a  wrought-iron  shaft. 

Because  of  the  near  equality  of  the  moduli  of  elasticity 
of  wrought  iron  and  steel,  the  deflection  and  torsional  angle 
of  wrought  iron  and  steel  under  stress  will  be  practically 
the  same  in  all  cases. 

(53.)  Journal-Bearings  of  the  Crank-Shaft. — In  the 

various  hand-books  of  mechanical  engineering,  giving  em- 
pirical rules  for  the  length  of  the  journals  of  shafts,  we  are 
advised  to  make  the  journal-bearing  from  1 J  to  2  times  the 
diameter  of  the  shaft.  In  Art.  (30),  formula  (81),  we  have 
the  means  of  determining  the  least  allowable  length  of  jour- 
nal of  shaft  under  the  most  favorable  circumstances — that 
is,  when  the  shaft  is  submitted  to  torsion  only,  and  does  not 


112 


THE  RELATIVE  PROPORTIONS 


bear  the  additional  weight  of  a  fly-wheel,  screw-propelloi 
or  paddle-wheels. 

If  to  the  stress  due  the  steam-pressure  on  piston  be  added 
the  weight  of  a  fly-wheel,  etc., 

Let  Q  =  reaction  at  bearing  due  to  weight. 
"    S  =  stress  due  steam-pressure  on  piston. 

Referring  to  Fig.  16,  we  see  that  the  force  S  always  acts 
in  the  direction  of  the  centre  line  of  the  cylinder,  and  that 
the  force  Q  acts  downward,  and  further  that  the  small  force 
S,  for  ordinary  lengths  of  connecting-rods,  causes  the  re- 


FlG.  16. 


sultant  R  of  the  forces  Q  and  S  to  vibrate  between  the  posi- 
tions and  values  R  and  Rl  when  the  crank  moves  in  the 
direction  of  the  arrow — i.  e.,  throws  over  and  tends  to  lift 
the  shaft  from  its  bearings.  If  the  crank  throws  under,  the 
small  force  Si  tends  to  press  the  shaft  down  upon  its  bearings. 
We  have  for  the  value  of  the  resultant  force,  neglecting  /Si, 

.R-l/tf+e^etfcoBa.  (149) 

For  the  angle  a  =  90  degrees — that  is,  for  a  horizontal 
engine — we  have 


OF  THE  STEAM-ENGINE.  115 

For  a  =  0  degrees  —  that  is,  for  a  vertical  engine  —  we  have 

jR-6+Sor  "I 

For  a  =  180  degrees,  we  have  >  .          (151) 

R=Q-S.  ) 

Let  R  =  pressure  on  journal  from  above  formulae. 
"    /  =  coefficient  of  friction. 
"    (£4  =  diameter  of  journal  of  shaft. 
"    /4  =  length  of  journal  of  shaft. 
"    W=  work  (or  heating)  allowed  per  square  inch  of  pro- 

jected area  per  minute  =  49908  inch-pounds. 
"    N=  number  of  strokes  per  minute. 

Then,  by  a  similar  course  of  reasoning  to  that  in  Art.  30, 
we  have 

(152) 


Example.  —  Let  us  take  a  horizontal  cylinder. 
Let  S=  60000  pounds. 
"  §  =  30000  pounds.     We  have 


R  -  V  Q* + £2  -  T/4500000000  -  approx.  6700  Ibs. 

Let  N=4Q  per  minute  and/=.08. 

Then,  by  formula  (152),  we  have 

/4  =  .0000325  x  .08  x  67000  x  40  =  6.9  inches, 
which  is  the  minimum  length  of  shaft-journal  allowable. 

The  importance  of  the  influence  of  the  number  of  turns 
upon  the  length  of  the  bearing  has  not  hitherto  been 
noticed,  and  the  use  of  empirical  rules  has  resulted  in  bear- 
ings much  too  long  for  slow-speeded  engines  and  too  short 
for  high-speeded  engines. 

To  cover  the  defects  of  workmanship,  neglect  of  oiling, 
and  the  introduction  of  dust,  it  is  probably  best  to  take 
/=  .16,  or  possibly  even  greater  if  we  make  use  of  formula  152. 

Five  hundred  pounds  per  square  inch  of  projected  area 
10* 


114  THE  RELATIVE  PROPORTIONS 

may  be  allowed  for  steel  or  wrought-iron  shafts  in  brass 
bearings  with  good  results,  if  a  less  pressure  is  not  attain- 
able without  inconvenience. 

Babbit  or  soft-metal  linings,  which  are  moulded  by  pour- 
ing in  the  metal  around  the  shaft  and  allowing  it  to  fit  in 
cooling,  are  used  in  some  forms  of  engine  with  great  economy 
and  good  results. 

For  great  pressures  the  shaft  is  sometimes  cased  in  gun- 
metal  and  the  casing  run  in  lignum-vitse  bearings,  which 
are  lubricated  with  water.  This  expedient  is  commonly 
used  for  propellor-shafts,  and  an  aperture  communicating 
with  the  hold  of  the  vessel  causes  a  constant  stream  of  water 
to  flow  through  the  bearing. 

Hollow  or  "  lantern "  brasses,  through  the  interior  of 
which  a  constant  stream  of  water  is  kept  flowing  in  order 
to  convey  away  the  heat,  are  also  used  for  great  pressures 
with  good  success. 

It  is  best,*  where  great  pressures  are  used,  to  have  some 
means  of  feeling  of  the  shaft  as  it  turns,  and  any  spot  which 
feels  rough  ("  ticklish ")  should  at  once  be  lubricated  by 
means  of  a  long-nosed  oil- can,  with  a  wick  in  the  end  of 
the  nose  placed  in  contact  with  it. 

By  means  of  these  expedients  pressures  of  1000  pounds  per 
square  inch  of  projected  area  have  been  successfully  used. 

With  very  slow  speed  even  greater  pressures  are  some- 
times used. 

Arthur  Rigg  gives  in  A  Practical  Treatise  on  the  Steam- 
engine  many  forms  of  plumber-blocks  and  bearings  from 
English  models. 

Warren's  Elements  of  Machine  Construction  and  Drawing 
gives  some  good  forms  of  the  French  and  American  types. 
Inspection  of  Fig.  16  shows  at  what  points  the  bearings  are 
liable  to  the  greatest  wear,  being  the  points  at  which  the 
resultant  E  intersects  the  circumference  of  the  bearings. 


OF  THE  STEAM-ENGINE.  115 


CHAPTER    XI. 

(54.)  Double  Cranks. — For  the  purpose  of  obtaining 
greater  regularity  of  revolution  of  the  crank-shaft,  as  well 
as  to  enable  the  engine  to  start  in  any  position,  two  cranks 
at  right  angles  are  frequently  used. 

We  shall  neglect  the  obliquity  of  the  connecting-rod,  and 
also  consider  the  pressure  upon  the  crank-pin  to  be  uniform 
throughout  the  stroke,  as  has  been  shown  possible  to  render 
it  approximately  in  Art.  (35). 


r/ 


Let  a  =  the  half  angle  between  the  two  cranks  I  and  II. 
"    0  =  the  variable  angle  B  C  O  formed  by  the  line  C  B, 

which  bisects  the  angle  I  C  II. 
"   $  =  the  force  acting  upon  each  crank-pin. 
"    r  =  the  radius  of  the  crank. 

Then,  Fig.  17,  we  have  for  the  combined  moments  of  the 
crank  =  y,  when  both  cranks  are  on  one  side  of  the  line  O  C, 

y  =  Sr[sin  (#-«)  +  sin  (#  +  «)]      (153) 
=  2£rsin  0  cos  «.  (154) 

Differentiating,  and  equating  with  0,  we  have 


at 


116  THE  RELATIVE  PROPORTIONS 

giving  a  maximum  for  6  =  90°  or  270°.     When  the  cranks 
C  I  and  C  II  are  on  opposite  sides  of  the  line  O  C,  we  have 

y  =  Sr  [sin  (0  -  a)  +  sin  (0  +  a  -  180)]        (155) 
-  2Sr  cos  d  sin  a.  (156) 

Differentiating,  and  equating  with  0,  we  have 


dO 

giving  a  maximum  for  0  =  0°  or  180°. 

Further,  it  will  at  once  be  seen,  if  we  consider  a  as  a  vari- 
able as  well  as  0,  that  the  value  of  expressions  (154)  and 
(156),  which  express  all  values  of  the  combined  moments, 
will  be  a  maximum,  and  therefore  the  possible  minimum 
value  of  the  expressions  differ  least  from  the  maximum 
values  if  we  make  sin  0  =  cos  «  and  cos  0  =  sin  «  —  a  condition 
which  can  only  be  fulfilled  by  letting  0  =  a  =  45°,  giving 
for  the  angle  between  the  cranks  90°,  and  for  the  position 
of  least  moment  6  =  45°  —  that  is,  when  one  of  the  cranks  is 
at  its  dead  point. 

We  have  then  for  a  maximum  value  of  equations  (154) 
and  (156) 

y  =  2Sr*  .7071  -  1.414ft-,  (157) 

and  for  a  minimum  value 

y  =  '2Srx.5  =  Sr.  (158) 

If,  as  in  the  case  of  a  marine-engine,  the  power  of  the 
first  crank  is  communicated  through  a  second  bent  crank, 
we  see  that  that  crank  and  the  shaft  leading  from  it  at  no 
time  sustain  twice  the  torsional  stress  exerted  by  one  crank, 
but  at  a  maximum  1.414  times  as  much  ;  and  in  determin- 
ing the  dimensions  of  the  after-crank  and  of  the  shaft,,  we 
should  regard  the  stress  as  that  upon  the  first  multiplied 
by  1.414. 


OF  THE  STEAM-ENGINE.  117 

Multiplying  the  numeri- 
cal   coefficient    of   equation  FIG.  18. 

(132)  by  i/lA\A,  we  have,  <* 

with  the  same  notation  for 
the  diameter  of  the  shaft 

(159) 


In  the  case  of  double  en- 
gines it  is  customary  to  make 

the  after-crank-pin  of  the  same  size  as  the  shaft,  for  the 
reason  that  it  is  subjected  to  many  unforeseen  stresses. 

The  length  of  the  after-crank-pin  should  be  the  same  as 
that  of  the  forward  pin.  See  Art.  (30). 

The  stress  upon  the  after-pin  due  to  its  cylinder  may  be 
regarded  as  constant,  and  in  the  direction  of  the  centre  line 
of  its  cylinder. 

The  stress  upon  the  after-pin  from  the  forward  crank-pin 
is  a  maximum  in  a  tangential  direction  to  its  circle  of  revo- 
lution, and  equal  to  S  when  the  forward  pin  makes  an  angle 
of  90°  with  its  cylinder  centre  line.  Fig.  18. 

If  we  make  the  supposition,  as  may  happen  to  be  the  case, 
that  both  these  stresses  act  simultaneously  at  the  forward 
end  of  the  after-pin  without  support  from  the  forward  crank,* 
we  have  for  the  maximum  stress  upon  the  pin  1/2$=  1.414$, 
and  for  the  diameter  of  the  after-pin,  from  formula  (84), 
Art.  (32), 

4  =  0.072^4^;  (160) 

or,  if  we  regard  the  steam-pressure  as  constant, 

(161) 


*  This  supposition  does  not  give  greater  results  than  are  often 
found  practically  necessary  by  an  expensive  tentative  process. 


118  THE  RELATIVE  PROPORTIONS 

It  should,  however,  be  remembered  that,  although  the 
most  unfavorable  suppositions  possible  for  the  known  stresses 
have  been  made  in  the  present  case,  unforeseen  stresses  are 
liable  to  occur,  which  can  be  guarded  against  only  by  making 
the  after-pin  the  same  size  as  the  shaft  if  possible. 

(55.)  Triple  Cranks. — Triple  cranks  120  degrees  apart 
are  sometimes  used  to  attain  still  greater  regularity  of 
motion  in  the  engine.  (Fig.  19.) 

Let  «  =  the  angle  made  by 
one  of  the  cranks 
with  the  line  AB. 
"   S  =  the  pressure  on  the 
piston-head  in  Ibs. 
"    r  =  the   radius    of   the 

crank  in  inches. 
We   have   for   the    tor- 
sional  moment  of  the  three 
cranks,  =  y, 

y  =  Sr[sin  a + sin(a + £TT)  +  sin(a + f  TT)],       (162) 

in  which  the  sines  are  taken  as  positive  because  the  cylinders 
are  double-acting. 

Further,  we  see  that  the  sum  of  the  sines  is  not  increased 

7T 

when  we  increase  each  of  them  by  -  =  60  degrees,  and  it  is 

3 

then  only  necessary  to  consider  the  equation  for  the  angle  a 

between  the  limits  0  and  -. 
3 

Reducing  equation  (162),  we  have 

y  =  Sr  [sin  a + l/3~cos  «].  (1 63) 

Neglecting  Sr,  differentiating,  and  placing  the  first  differ- 
ential coefficent  =  0,  to  find  the  maximum  and  minimum 
values,  we  have 


OF  THE  STEAM-ENGINE.  119 

—  =  cos  a  -  l/3sin  a  =  0, 
da 

~-  =  -  sin  a  - 1/3  cos  a, 
da* 

giving  for  a  maximum  value 

cos  a  =  1/3  sin  a. 

Therefore,  tan  a ;  therefore,  a  =  30°  =  -. 

1/3  6 

The  minimum  values  of  equation  (163)  occur  when  a  =  0 
and  =  -  =  60°. 

o 

We  see  that  that  part  of  the  shaft  attached  to  the  third 
crank  is  subjected  to  a  maximum  torsional  stress  twice  as 
great  as  that  due  to  one  cylinder.  Its  diameter  can  be  most 
readily  calculated  by  doubling  the  actual  steam-pressure  in 
equation  (132). 

In  the  same  manner,  the  proper  proportions  of  the  crank 
can  be  calculated  by  Art.  (37). 

In  the  case  of  three  cranks,  in  order  to  find  the  maximum 
stress  to  which  the  second  crank  and  shaft  may  be  submitted, 
we  must  find  the  maximum  of  the  expression 

/      2   \] 
sin  «  +  sin  j  a+-7r  )  (164) 

Reducing,  and  neglecting  &r, 

y  =  -  sin  a  +  1L_  cos  a. 

Differentiating,  and  placing  =  0,  we  have 

dy    3  j/3". 

—  =  -  cos  a  -  ^—  sin  a  =  0. 
da     2  2 


120  THE  RELATIVE  PROPORTIONS 

Therefore  tan  a  =  T/37  therefore  a  =  60°  =  -, 

o 

gives  the  greatest  maximum  to  which  they  are  subjected, 
and  equation  (164)  becomes 

2/  =  £r[0.866  + 0.866]  =  1.732  /Sr, 

which  is  also  the  minimum  value  of  the  torsional  stress  for 
3  cranks  together. 

We  can  calculate  the  proper  proportions  of  the  crank, 
si] aft  and  crank-pin  by  multiplying  the  steam-pressure  by 
1.732  in  equation  (132). 

(56.)  The  Fly- Wheel.— Before  taking  up  the  subject  of 
the  fly-wheel  mathematically,  it  will  perhaps  be  best  to  give 
a  general  idea  of  its  function. 

It  is  impossible  to  control  the  speed  of  any  engine  for  any 
considerable  length  of  time  by  means  of  a  fly-wheel,  or  to 
render  the  motion  of  any  engine  exactly  uniform  for  any 
period  of  time. 

It  is,  however,  possible,  by  properly  proportioning  the 
weight,  diameter  and  speed  of  rim  of  a  fly-wheel  to  the 
work  given  out  by  the  steam-cylinder,  to  confine  the  varia- 
tion of  the  speed  of  the  engine  from  any  assigned  mean 
speed  during  the  time  of  one  stroke,  within  any  assigned 
limits. 

A  fly-wheel  serves  to  store  up  work,  or  to  give  it  out  when 
required,  just  as  a  mill-pond  fed  by  a  stream  of  variable  dis- 
charge serves  to  store  up  water  for  the  mill-wheel ;  the  larger 
the  pond,  the  less  the  effect  upon  it  of  any  sudden  increase 
or  diminution  of  the  water  flowing  into  it,  and  so  with  the 
fly-wheel :  the  larger  it  is,  and  the  more  rapid  its  motion,  the 
more  steadily  it  will  run,  so  that  it  would  hardly  be  possible, 
where  a  uniform  motion  in  one  direction  only  is  desired,  to 
make  a  fly-wheel  too  large,  were  it  not  for  the  circumstances 
that  the  loss  of  work  due  to  increased  friction  and  its  greater 
cost  limit  us  in  that  direction. 


OF  THE  STEAM-ENGINE.  121 

In  considering  the  weight  and  speed  of  a  fly-wheel,  that 
only  of  its  rim  will  be  considered,  and  it  is  also  proper  to 
state  here,  to  avoid  leading  our  readers  astray,  that,  unless 
specially  noted,  the  formulae  to  be  subsequently  established 
do  not  take  cognizance  of  the  variation  in  work  given  out  by 
the  steam-cylinder  produced  by  the  angular  position  of  the 
connecting-rod  ;  that  the  suppositions  are  also  made  that  the 
mean  pressure  of  the  steam  is  uniform  throughout  the  stroke ; 
and  that  the  work  given  out  by  the  fly-wheel  is  given  out 
uniformly. 

The  variation  in  the  work  given  out  by  the  steam-cylin- 
der, produced  by  the  angularity  of  the  crank,  is  specially 
considered. 

When  any  body  of  a  weight  W  is  in  motion  with  a  ve- 

TFv2 
locity  v,  it  has  stored  up  in  it  work  =  w  = ,  and  all  of 

this  work  must  be  given  out  before  it  can  come  to  rest.  If 
this  weight  is  not  entirely  brought  to  rest,  but  its  speed  re- 
duced to  »!,  it  will,  while  being  retarded,  give  out  work 


(165) 


Or  if  the  body  be  moving  with  a  velocity  v1}  and  by  the 
action  of  a  force  its  speed  be  increased  to  a  velocity  v,  it  will  ' 
store  up  work 


Now,  (vj  —  Vi2)  =*(v+Vi*)(v  —  v^),  and  if  we  take  the  mean 
uniform  speed  of  the  body  =  u  = -,  when  v  and  Vi  are 

supposed  not  to  differ  greatly,  and  denote  by  m  the  frac- 
11 


122  THE  RELATIVE  PROPORTIONS 

tional  part  of  the  mean  velocity  u,  by  which  v  and  Vi  are 
allowed  to  differ,  we  have  mu  =*v  —  v1}  and  we  have 


(v2  -  Vj2)  =  (v  +  Vi~)  (v  -  vj 
and  formula  (165)  takes  the  following  form  : 

w-m™*.9  (166) 

9 

in  which  ^  =  32.2,  and  which  gives  the  amount  of  work 
gained  or  lost  by  the  body  when  its  speed  is  increased  or 
decreased  by  (v  -  1^)  =  mu. 

For  the  present  it  will  suffice  to  say  that  m  is  taken  from 
i  to  TOIT>  according  to  the  degree  of  regularity  required,  and 
that  u  is  the  mean  speed  per  second  in  feet. 

We  will  next  take  up  the  amounts  of  work  lost  and 
gained  by  the  fly-wheel  while  receiving  work  from  the 
steam-cylinder  in  periodically  varied  quantities,  and  part- 
ing with  it,  on  the  other  hand,  in  uniform  quantities  to  the 
machinery  driven. 

If  the  work  is  not  given  out  in  uniform  quantities,  as  is 
sometimes  the  case,  special  provision  should  be  made  for  it 
by  increasing  the  weight  of  the  fly-wheel,  so  as  to  meet  and 
overcome  this  source  of  irregularity,  or,  better,  by  the  use  of 
a  fly-wheel  at  the  point  at  which  the  variations  occur,  cal- 
culated to  meet  and  overcome  the  irregularities  at  that 
point. 

"With  an  early  cut-off  of  the  steam,  the  irregularity  due 
to  the  variable  pressure  on  the  piston  is  superadded  to  that 
due  to  the  angular  position  of  the  crank  and  the  connecting- 
rod,  and  will  be  specially  noted  hereafter  in  Table  IV.  for 
such  cases  as  may  occur  in  which  the  weight  of  the  piston 
and  appurtenances,  is  not  or  cannot  be  proportioned  to  the 
speed  of  the  reciprocating  parts  and  the  steam-pressure. 
See  Art.  (35). 


OF  THE  STEAM-ENQINE.  123 

To  establish  a  clear  understanding  between  the  reader 
and  ourselves,  we  here  define  work  to  be  force  multiplied  by 
the  space  passed  over  by  that  force. 

(57.)  Fly-  Wheel,  Single  Crank.—  Let  JP=the  steam- 
pressure  per  square  inch  upon  the  piston-head. 

Let  d  =  the  diameter  of  the  steam-cylinder  in  inches. 
"    /S^the  total  pressure  upon  the  piston-head. 

The  force  pressing  upon  the  piston-head  is,  as  before 
stated,  for  a  single  cylinder, 

S  =  ^P.  (167) 

4 

(Assuming  the  engine  to  have  attained  its  average  speed, 
and  neglecting  the  small  variation  produced  by  the  connect- 
ing-rod,) 

Letting  r  =  radius  of  crank, 

"       a  =  angle  formed  by  centre  line  of  crank  with  the 

centre  line  of  cylinder, 
"       s  =  space  passed  over  by  the  piston, 

the  distance  passed  over  by  the  piston  is 

s  =  r(l-cos«).  (168) 

Therefore,  we  have  for  the  work  derived  from  the  pis- 

ton =  Wi 

cosa).  (169) 


To  find  the  increments  of  work  corresponding  to  each 
increment  of  arc,  we  differentiate  (169),  giving 

dw^Srsin  ada.  (170) 

The  total  amounts  of  work  gained  and  lost  by  the  fly- 
wheel during  one  stroke  equal  SL.  \_L  —  the  length  of  the 
stroke.] 


124  THE  RELATIVE  PROPORTIONS 

As,  however,  the  work  is  gained  in  varied  increments,  as 
shown  by  (170),  and  on  the  other  hand  is  lost  in  assumed 
uniform  quantities  to  the  machinery  driven,  there  are  points 
at  which  the  increments  of  the  gained  and  the  lost  work  are 
equal. 

The  lost  work  during  the  passage  of  the  crank  through 
the  angle  a  is 

ra.  (171) 


8 

Since  =  the  lost  work  for  the  unit  of  arc,  and  differ- 
entiating (170),  we  have 

dw^-Srda.  (172) 

7T 

Placing  equations  (170)  and  (172)  equal  to  each  other, 
we  have 

2 
Sr  sin  a  =  -  Sr. 

7T 

Therefore,  sin  a  -  -  -  0.636618,          (173) 

7T 

which  gives  the  following  values, 

Angle  «  -  39°  32' 25" 
or  =  140°  27' 35" 
or  =  219°  32' 25" 
or  =  320°  27'  35" 

We  see  from  equation  (170)  that  the  increment  of  gained 
work  is  also  equal  to  0  for  a  =  0°  or  180°,  and  a  maximum 
for  a  =  90°  or  270°,  while  the  increment  of  lost  work,  equa- 
tion (172),  is  a  constant. 

If  now  we  substract  the  value  of  equation  (169)  for 


OF  THE  STEAM-ENGINE.  125 

a  -  39°  32'  25"  from  its  value  for  a  - 140°  27'  35",  we  have 

Wi  =  $r[(l  +  cos  a)  -  (1  -  cos  a)]  ; 
therefore,        w,  =  2Sr  cos  a  -  1.54232£r,  (175) 

and  (175)  is  the  total  amount  of  work  received  from  the 
steam-cylinder  by  the  fly-wheel  from  the  point  where  the 
increment  of  gained  work  is  equal  to  the  increment  of  lost 
work  until  they  are  again  equal.  In  order  to  find  the  sur- 
plus of  work  absorbed  by  the  fly-wheel,  we  must  subtract  the 
total  amount  of  lost  work  (lost  uniformly)  during  the  same 
interval. 

If  now  we  subtract  the  value  of  equation  (171)  for 
a  =  39°  32'  25"  from  its  value  for  a  - 140°  27'  35",  we  have 

100°.91945  \ 

IT: 

180°       / 

Substracting  (176)  from  (175),  we  have  for  the  work 
stored  by  the  fly-wheel  between  the  two  values  of  a,  above 

stated,  =  i03, 

W3  -  (Wl  _  w2)  =  0.42099£r, 

and  substituting  for  r  its  value  — -,  we  have,  for  a  single 

2 

cylinder, 

fK,-.2104aSL,     *  (177) 

or  about  .21  of  the  work  done  during  the  whole  stroke. 

If  we  substitute  for  S  its  value  given  in  equation  (167), 
and  take  L  in  feet,  we  have,  in  terms  of  the  pressure  diam- 
eter of  steam-cylinder,  letting  c  represent  the  numerical 
coefficient  derived  from  calculation  or  Table  IV., 

i03  =  .7854P<f  cL.  (178) 

To  prove  that  the  work  lost  by  the  fly-wheel,  in  passing 
from  crank-angle  140°  27'  35"  to  219°  32'  25",  is  equal  to 
11* 


126  THE  RELATIVE  PROPORTIONS 

that  gained  in  passing  from  crank-angle  39°  32'  25"  to 
140°  27'  35". 

Referring  to  equation  (175),  we  see  that  the  total  work 
gained  between  the  limits  a  =  140°  27'  35"  and  a  =  219°  32' 
25",  if  we  properly  alter  the  signs,  is 


and,  referring  to  equation  (171),  proceed  as  in  equation  (176), 
we  have  for  the  total  lost  work  of  the  fly-wheel 

=  2^(0.4393).       (180) 


cos  a  =  0.77116,     .  •  .  (1  -  cos  a)  =  .22884. 
Subtracting  equation  (179)  from  (180),  we  have 
(wt  -  «0  =  2$r(0.4393  -  .2288)  -  .421  Sr  =  w3  the  work  lost, 

and  we  see  that  the  amount  of  work  gained  by  the  fly-wheel 
in  an  arc  of  100.92  degrees  is  equal  to  that  lost  in  an  arc 
of  79.08  degrees. 

Now,  for  the  sake  of  a  clear  understanding  of  this  topic, 
let  us  follow  the  crank  through  one  revolution.  (See  Fig.  20.) 
From  crank-angle  39°  32'  25"  to  140°  27'  35"  the  an- 
gular velocity  of  the  crank  and  fly-wheel  increases,  attaining 
FIG  ^  a  maximum  at  140°  +  ,  the  fly- 

wheel storing  up  work.     From 

140°  +  to  219°  +  the  anSular  ve~ 
locity  of  the  crank  and  fly- 

wheel decreases,  reaching  a  min- 
imum  at  219°  +  ,  the  fly-wheel 
giving  out  work  equal  in  amount 
to  that  stored  up  in  the  first  arc 
mentioned  ;  from  219°  +  to  320°  + 

the  fly-wheel  again  stores  work,  and  from  320  °+  to  39°  + 

gives  the  same  amount  up. 


OF  THE  STEAM-ENGINE.  127 

(58.)  Fly-Wheel,  Double  Crank,  Angle  90°.—  The 

most  advantageous  angle  between  the  cranks  being  90°,  as 
shown  in  Art  (54),  the  total  work  of  the  two  cranks  will  be, 
if  we 

Let  a.  =  the  angle  between  the  centre  line  of  the  cylinder  and 

the  first  crank, 

"   S  =  the  pressure  in  pounds  upon  each  piston-head, 
"    r  =  the  radius  of  the  crank, 
"  M!  =  the  work  given  out  by  the  two  cylinders, 

since  cos  (90  +  a)  =  —  sin  a, 

-l)].         (181) 


Reducing  and  differentiating,  we  have  the  increment  of 
gained  work  for  each  increment  of  arc 

dwi  =  /Sr(sin  a  +  cos  a)da.  (182) 

Multiplying  equation  (172)  by  2,  we  have  for  the  incre- 
ment of  the  lost  work  for  each  increment  of  arc 

dw^-Srda.  (183) 

7T 


Equating  equations  (182)  and  (183),  we  have 

4 
Sin  a  +  COS  a  =  -. 

7T 

Squaring  both  members, 

/4\ 
sin2  a  +  cos2  a  +  2  sin  a  cos  «  =  [-]. 

w 

Therefore,  sin  2«  -  (*  Y  - 1  -  0.62114.     (184) 

Giving  2a  -  38°  24' 

and  «  =  19°  12'. 

Remembering  that  after  passing  through  90°  of  arc  the 


128  THE  RELATIVE  PROPORTIONS 

same  variations  are  repeated,  we  have  for  the  various  values 
of  a,  when  the  increments  of  lost  and  gained  work  are  equal, 

19°  12'..  ...51°  36' 


109°  12'  51o  36, 

160°  48'  ' 

199°  12' 

250°  48' 

289°  12T 

340°  48',  etc. 

From  (182)  we  further  see  that  the  increment  of  gained 
work  is  a  maximum  for  a  =  45°  and  a  minimum  for  a  =  0° 
or  90°. 

To  determine  the  excess  of  work  lost  or  gained,  we  have, 
equation  (181)  reduced, 

Wi  =  $r[l  —  cos  a  -i-  sin  a]. 

For  a  =  70°  48'  this  equation  ....................  =Sr*  1.61  5509 

For  a  =  19°  12'  this  equation  ....................  =£rx  0.384491 

The  total  amount  of  work  gained  ...........  =  Sr*  1.231018 

K"1    O/J 

The  total  amount  of  work  lost  is  4Sr  —  -  —  =  Sr  x  1.146667 

180 
Giving  for  the  gained  or  lost  work  ............  Sr  x  0.084351 

If  for  r  we  substitute  its  value  —  ,  we  have 

w,-0.0422SL.  (185) 

As  in  the  preceding  article,  we  see  that  the  angles  deter- 
mined correspond  to  points  of  minimum  and  maximum  an- 
gular velocity  of  the  fly-wheel,  and  it  is  easy  to  prove  that 
the  work  gained  by  the  fly-wheel  in  passing  from  say  19°  12' 
through  51°  36'  is  equal  to  the  work  lost  in  passing  from 
70°  48'  through  an  arc  of  38°  24'. 


OF  THE  STEAM-ENGINE.  129 


CHAPTER    XII. 

(59.)  Fly-Wheel,  Triple  Crank,  Angle  120°.— Let- 
ting the  notation  be  the  same  as  in  the  preceding  article,  we 
have,  for  the  work  given  out  by  the  three  cranks, 

Wi  =  >Sr[(l  -  COS  «)  +  (!-  COS  (a  +  Jrr)  -  f ) 

+  l-cos(«+f7r)-i)],  (186) 


s 


which,  after  reduction,  becomes 

W=  Sr[l  -  cos  a  +  i/Fsur  a],  (187) 

Differentiating,  and  neglecting  the  constants,  we  have 


(188) 
da 

n 

which  can  be  placed  equal  to  -  =  the  work  lost  in  each  ele- 

ment of  arc,  to  find  the  points  at  which  the  increments  of 
the  gained  and  the  lost  work  are  equal. 


-|/3~cos  a  =  sin  a  cos  60°  +  cos  a  sin  60 
since  -  -j/Fand  cos  60°  =  J. 


130  THE  RELATIVE  PROPORTIONS 

We  then  have 

sin  (a  +  60°)  =  -. 

7T 

Therefore,  («  +  60)  =   72°  44'  and  a  =  12°  44', 
or          (a  +  60)  =  107°  16'  and  a  =  47°  16'. 

Since  the  same  phases  are  repeated  after  the  cranks  have 
passed  through  an  angle  of  60  degrees,  we  need  consider 
this  equation  only  between  0  and  60  degrees. 

Expression  (188)  becomes  a  minimum  for  «  =  0  or  =  60°, 
and  is  a  maximum  for  «  =  30°. 

To  determine  the  amount  of  work  lost  or  gained  by  the 
fly-wheel,  we  substitute  in  equation  (187)  the  values 
a  =  47°  16'  and  a  =  12°  44'. 

For  «  =  47°  16'  equation  (187)  becomes  ..........   £rxl.5936 

For  a  =  12°  44'  equation  (187)  becomes  .........  .   Sr*  0.4063 

The  total  amount  of  work  gained.  is  ...............    $rx  1.1873 

The  total  amount  of  work  lost  is  for  an  arc  | 

of  34°  32'=34.°533  .........  :  ...............  }_ 

Giving  for  the  gained  work  absorbed  by  the  "> 

fly-wheel  in  an  arc  of  34.°533  .............  } 

To  find  the  work  lost  by  the  fly-wheel  as  a  check,  we  have 

For  the  work  lost  in  an  arc  of  25°  28'  ............  $rx  0.8488 

For  the  work  gained  in  an  arc  of  25°  28'  .........  Srx  0.8126 

Giving  for  the  lost  work  given  out  by  the  fly-  ) 

wheel  in  an  arc  of  25°  28'  ..................  j  ^rxO.0362 

If  for  r  we  substitute  its  value  —  ,  we  have 

2 


(189) 

"We  can,  in  a  similar  manner  to  that  already  explained, 
determine  the  six  points  of  maximum  and  the  six  points  of 
minimum  velocity  of  the  fly-wheel. 


OF  THE  STEAM-ENGINE. 


131 


(60.)  Of  the  Influence  of  the  Point  of  Cut-off  and  the 
Length  of  the  Connecting-Rod  upon  the  Fly-Wheel.— 

Let  $  =  the  total  pressure  of  the  steam  upoa  the  piston-head. 
"    r  =  radius  of  crank. 
"     l  =  nr  =  the  length  of  the  connecting-rod. 
"    a  =  the  angle  between  the  crank  and  the  centre  line  O  X. 
"    <p  =  the  angle  between  the  connecting-rod  and  the  cen- 
tre line  OX.     (See  Fig.  21.) 
FIG.  21. 

r 


Referring  to  equation  (170),  Art.  (37),  we  see  that  the 
increments  of  work  given  out  with  a  varying  velocity  are 
proportional  to  sin  a,  the  force  S  being  assumed  constant ; 
as  the  crank-pin  may  be  assumed  to  move  in  a  tangential 
direction  with  a  constant  velocity,  the  condition  of  the 
equality  of  the  increments  of  work  at  the  points  O  and  B  * 
requires  that  the  tangential  or  torsional  force  T  shall  vary 
as  the  sin  a.  (This  can  be  proved  graphically  also.) 

CASE  I. — Let  S  be  constant  and 
the  length  of  the  connecting-rod 
be  assumed  infinite. 

With  a  radius  AB,  Fig.  22, 
representing  the  constant  force  S 
acting  on  the  piston  to  a  con- 
venient scale,  describe  the  semi- 
circle A  3  C.  Divide  this  semi-perimeter  into  any  number 

*  Theorem  of  virtual  velocities. 


FIG.  22. 


1 


132 


THE  RELATIVE  PROPORTIONS 


FIG.  23. 


of  equal  spaces,  as  1,  2,  3,  4,  5,  C.  From  these  points  drop 
verticals  to  the  line  AC;  these  verticals  will  represent  the 
tangential  forces  T  at  these  points. 

If  we  lay  off  a  horizontal  line,  as  O  X,  Fig.  23,  representing 
the  semi-perimeter  of  the  crank-circle  to  any  convenient 
scale,  and  divide  it  into  six  equal  parts,  and  upon  these 
divisions  erect  verticals  of  equal  length  to  the  verticals  in 

Fig.  22,  we  obtain  the 
curve  of  sines  O  D  6,  and 
the  area  of  this  figure, 
O  D  6  O,  represents  the 
work  done  in  one  stroke. 

u      '      *     J     *      *     °    *   The  work  lost  by  the  en- 
gine,  being  assumed  lost 

uniformly,  can  be  represented  by  the  rectangle  O  E  G  X  O, 
in  which  the  vertical  OE  =  G6  =  the  work  done  in  one 
stroke  divided  by  the  distance  O  6,  and  the  sum  of  the 
areas  of  EHO  +  KG6  =  area  DK H. 

CASE  II. — Let  &  be  variable  (steam  cut-off)  and  the 
length  of  the  connecting-rod  assumed  infinite. 

'  The  different  pressures  caused  by  cutting  off  steam  can 
be  calculated  by  Mariotte's  law,  or  more  correctly  taken 
from  an  indicator  diagram,  as  shown  in  Fig.  24,  by  laying 
off  versines,  say  for  each  30  degrees  of  arc  on  a  diameter 
equal  to  the  length  of  the  indicator  diagram. 
FIG.  24. 


V 


With  the  greatest  force  as  a  radius  =  O  a,  Fig.  24,  describe 


OF  THE  STEAM-ENGINE. 


133 


the  semi-circle  A  3  C  with  3  as  a  centre,  Fig.  25 ;  divide  this 
into  six  equal  arcs  and  draw  the  radii  1,  2,  3,  4,  5  to  3; 
upon  these  radii  lay  off  the  forces  as  measured  from  the  cor- 
responding ordinate  of  the  indicator  diagram,  Fig.  24.  Con- 
necting these  extremities  we  have  the  curve  A  1,  2,  3,  4j,  51} 
61,  and  the  verticals  11,  22,  33,  41;  4,  5^  5  ;  from  the  intersec- 
tions of  this  curve  with  the  radii  to  the  line  A  C  give  the 
tangential  forces  acting  upon  the  crank. 

FIG.  26. 


O       / 


If  we  divide  the  line  O  X,  Fig.  26,  representing  the  length 
of  half  the  crank-circle  to  any  convenient  scale,  into  six  equal 
parts,  and  at  the  points  of  division  erect  verticals  equal  in 
length  to  the  verticals  in  Fig.  25,  we  obtain  the  irregular 
solid  curve  ODX.  The  area  of  this  figure,  ODXO, 
equals  the  work  done  in  one  stroke. 

CASE  III. — Let  the  force  S  be  variable  and  the  length 
of  the  connecting-rod  be  taken  into  consideration. 

FIG.  21. 


Referring  to  Fig.  21,  we  see  that  the  effect  of  the  connect- 
12 


134  THE  RELATIVE  PROPORTIONS 

ing-rod  of  finite  length  is  to  cause  the  piston  to  move  farther 
than  the  crank-pin  in  a  horizontal  direction  in  the  first  and 
fourth  half  strokes,  and  to  move  a  less  distance  than  the 
crank  in  a  horizontal  direction  in  the  second  and  third  half 
strokes.  This  can  be  shown  in  the  following  manner : 

We  have,  for  the  work  done  in  the  cylinder  with  a  vari- 
able velocity  and  on  the  crank-pin  with  a  constant  velocity, 

w  =  £r[(l  -  cos  «) +n(l  -"cos  ?)]  J  (190) 

.  2        sin2  a     . 
sm2  <p  =  — — ,  since  r  sin  a  =  nr  sin  <p  ; 


/Sri  -cos  «  + 


sin 


2n 


Neglecting  quantities  containing  greater  than  the  second 

power  of  -  ,  and  differentiating,  we  have 
%  n 


«       o 

-  —  or  tern  a  +  —  2  sin  a  cos  a\  , 
da.  2ra 

and  the  tangential  force  which  is  proportional  to  the  first 
differential  coefficient  of  the  work,  since  the  velocity  of  the 
crank-pin  is  constant,  is  proportional  to 

sin2a  /ini\ 

sin  a+  --  .  (191) 

2n 

Attention  must  be  paid  to  the  signs  of  the  circular  func- 
tions. 

Laying  down  a  straight  line  O  X,  Fig.  26,  dividing  it, 
and  erecting  ordinates  as  before,  we  lay  off  the  length  of 


OF  THE  STEAM-ENGINE. 


135 


these  ordinates,  which  in  the  first  quadrant  are  greater  and 

in  the  second  quadrant  less  by than  for  the  solid 

curve  O  D  X. 


FIG.  26. 


In  the  third  quadrant  the  ordinates  will  be  less  and  in 

the  fourth  quadrant  greater  by than  in  Case  II., 

£11 

giving  the  broken  line  curve  O  b  c  d  e  6. 

The  work  lost,  being  lost  uniformly,  can  be  represented  by 
the  rectangle  O  E  G  X  O,  and  the  height  of  this  rectangle 
=  O  E  =  G  X  is  equal  to  the  work  done  in  one  stroke,  repre- 
sented by  the  figure  O  b  D  d  X  O,  divided  by  half  the 
length  of  the  crank-circle  =  O  X. 

The  areas  of  these  figures,  as  also  the  equal  amounts  of 
work  lost  and  gained  by  the  fly-wheel,  can  be  calculated  by 
means  of  Simpson's  rule,  or  more  conveniently  by  the  use 
of  the  polar  planimeter. 

Table  IV.  gives  the  work  lost  and  gained  by  the  fly-wheel 
in  terms  of  the  fractional  part  of  the  whole  work  done  by  one 
cylinder  in  one  stroke,  or,  what  is  the  same  thing,  the  value 
of  the  coefficient  c  in  equation  (178)  for  the  common  values  of 

n  =  -,  and  the  usual  points  of  cut-off  when  the  influence  of 

the  weight  and  velocity  of  the  reciprocating  parts  is  dis- 
regarded. 

If  we  wish  to  use  this  table  when  the  indicated  horse-power 
of  an  engine  is  given,  divide  the  horse-power  of  one  cylinder  by 


136 


THE  RELATIVE  PROPORTIONS 


the  number  of  strokes  per  minute,  and  multiply  it  by  33000 
foot-pounds,  since  the  work  done  in  one  stroke 


33000 


and  multiply  the  result  by  the  coefficient,  given  in  the  table. 

TABLE  IV. 
[From  Des  Ingenieurs  Taschenbuch,  page  379.] 


Engine  without 
expansion. 

Engine  with  expansion. 
Steam  cut  off  at- 

I 

Single 

Two 

Three 

4L 

* 

1Z» 

4Z, 

j£ 

*L 

\L 

* 

n>  =  r 

crank. 

cranks. 

cranks. 

2 

3-^ 

* 

7 

4 

0.2717 

0.1672 

0.0693 

(f  Single 
1  crank 

0.3741 

0.4076 

0.4372 

0.4523 

0.4625 

0.4702 

5 

0.2577 

0.1422 

0.0594 

6 
7 

0.2489 
0.2489 

0.1256 
0.1136 

0.0504 
0.0453 

(Two 
(cranks 

0.2044 

0.2250 

0.2412 

0.2495 

0.2552 

0.2594 

8 

0.2384 

0.1046 

0.0414 

/Single 
M  crank 

0.3252 

0.3594 

0.3916 

0.4088 

0.4216 

0.4300 

Infinite 

0.2105 

0.0422 

0.0181 

4 

\f  Two 
(cranks 

0.0652 

0.0720 

0.0786 

0.0820 

0.0846 

0.0862 

It  is  the  best  plan  to  make  use  of  the  coefficient  of  single 
cranks  for  double  cranks,  and  of  the  coefficient  for  double 
cranks  for  treble  cranks,  since  it  is  sometimes  necessary  to 
disconnect  one  cylinder  for  repairs,  which  we  should  be  able 
to  do  without  creating  such  irregularities  as  will  render  the 
engine  unserviceable. 

If  we  take  the  pressures  upon  the  crank-pin  as  altered  by 
the  weight  and  velocity  of  the  reciprocating  parts,  as  shown 
in  Fig.  10,  Art.  35,  and  treat  them  as  explained  in  this 
article,  we  can  obtain  an  exact  representation  of  the  work 
lost  and  gained  by  the  fly-wheel.  For  ordinary  practice, 
with  the  weight  and  velocity  of  the  reciprocating  parts 
properly  adjusted,  we  can  use  the  coefficients  for  engines 
without  expansion  in  Table  IV. 


OF  THE  STEAM-ENGINE.  137 


CHAPTER    XIII. 

(61.)  The  Weight  of  the  Rim  of  Fly-Wheels.— 

Referring  to  equation  (166),  we  see  that  the  amount  of  work 
which  any  body  can  store  up  without  having  its  velocity  in- 
creased, or  give  out  without  having  its  velocity  decreased 
more  than  the  fraction  m  of  the  mean  velocity  u,  is 

w  =  —  Wu\ 
9 

If  now  we  place  this  amount  of  work,  equal  to  the  amount 
given  by  equation  (178),  as  the  excess  or  deficiency  of  work 
to  be  stored  or  given  out  by  the  fly-wheel,  we  have 

in  .^ 

9     ' 

therefore,  W,=  25.29—    ±^,  (192) 

m        v? 

in  which  L  is  the  length  of  stroke  in  feet,  and  u  the  mean 
velocity  of  the  rim  of  the  wheel  in  feet  per  second. 

If  for  u  we  wish  to  substitute  the  mean  diameter  of  the 
fly-wheel  rim  D  and  the  number  of  strokes  per  minute  N, 


-.  02618  DJV. 


2x60 
Squaring  and  substituting  in  formula  (192),  we  have 

(193) 


The  mean  diameter  of  a  fly-wheel  D  is  usually  taken  at 
from  3  to  10  times  the  length  of  the  stroke.     Inspection 
12* 


138  THE  RELATIVE  PROPORTIONS 

of  formula  (193)  shows  that,  other  things  being  equal,  it 
will  diminish  the  weight  and  cost  of  a  fly-wheel  to  increase 
the  number  of  strokes  the  diameter  of  the  wheel,  or  to  in- 
crease the  fractional  coefficient  m. 

(62.)  Value  of  the  Coefficient  of  Steadiness  m. 
— Rankine  recommends  taking  m  =  -fa  for  ordinary  ma- 
chinery, and  = -fa  to  fa  for  machinery  requiring  unusual 
steadiness. 

Watt's  rule,  given  by  Farey,  and  regarded  by  him  as 
giving  sufficient  regularity  for  the  most  delicate  purposes,  is, 
"  Make  \  the  vis  viva — i.  e.,  the  work — stored  in  the  fly-wheel 
equal  to  the  work  done  by  the  engine  in  3f  strokes,"  and 
gives  m  =  -fa. 

Bourne's  rule  for  all  engines  is,  "  Make  the  work  stored 
in  the  fly-wheel  equal  to  that  developed  by  the  steam- 
cylinder  in  six  strokes,"  which  would  give  m  little  greater 
than  -fa,  a  quantity  much  too  small  for  ordinary  purposes, 
but  nearer  right  than  he  is  usually.  Nystrom  suggests 
making  m,-  in  practice,  to  vary  between  -fa  and  yjg-. 

Morin  gives,  for  engines  requiring  great  regularity,  the 
value  of  m  at  from  -fa  to  ^,  which  conforms  with  the  best 
practice,  for  engines  of  great  steadiness  of  motion. 

m  =  -fa  is  a  good  value  for  engines  in  which  small  tem- 
porary fluctuations  of  speed  are  of  little  consequence. 

The  following  values  of  m  are  taken  from  Des  Ingenieurs 
Taschenbuch,  Hiitte,  page  378 : 

For  machines  which  will  permit  a  very  uneven  motion,  as 
for  hammer-work,  m  =  £. 

For  machines  which  permit  some  irregularity,  as  pumps, 
shearing-machines,  etc.,  m  =  -fa  to  -fa. 

For  machines  which  require  an  approximation  to  a  uni- 
form speed,  as  flour-miDs,  m  =  -fa  to  fa. 


OF  THE  STEAM-ENGINE.  139 

For  machines  which  demand  a  tolerably  uniform  speed, 
as  weaving-machines,  paper-machines,  etc.,  m  =  -£$  to  -fa. 

For  machines  which  demand  a  very  uniform  speed,  as 
cotton-spinning  machinery,  m  =-  -fa  to  ^j. 

For  spinning  machinery  for  very  high  yarn  numbers, 


Example.  —  Let  n  =  -  =  5  ;  then  from  column  first  of  the 
r 

table  we  have  c  =  .2577  =-  \  approx. 

Let  the  uniform  pressure  be  P=61.5  pounds  per  square 
inch,  ro—jfo,  i-1  foot,  d  =  12  inches,  D  =  5  feet,  JV-lOO 
strokes  per  minute. 

Substituting  in  formula  (193),  we  have 

^     36899x20x61.5x144 
Wf=  -  -  ---  =  6535  pounds. 
4x25x10000 

Upon  reflection,  we  see  that  this  would  be  comparatively 
a  very  great  weight,  and,  if  possible,  it  will  be  best  to  in- 
crease the  diameter  of  the  fly-wheel,  so  as  to  lessen  its  weight. 
Letting  D  =  10  feet,  we  have 

Wf=  1634  pounds, 
with  an  equal  coefficient  m  of  steadiness. 

(63.)  Area  of  the  Cross-Section  of  the  Rim  of  a  Fly- 
Wheel.  —  Given  the  weight  and  mean  diameter  of  ike  rim  of 
a  fly-wheel  to  determine  the  area  of  its  cross-section. 

Let  D  be  the  mean  diameter  in  feet  of  a  fly-wheel  rim. 
"  Wf  be  the  weight  in  pounds. 

"    F  be  the  cross-section  of  a  fly-wheel  rim  in  square  inches. 
We  have  for  the  mean  perimeter  in  inches 

127TD, 


140  THE  RELATIVE  PROPORTIONS 

and  for  the  volume  of  the  rim  in  cubic  inches 


One  cubic  inch  of  iron  weighs  about  .26  of  a  pound,  and 

we  therefore  have 

TF7=.  26x12^1). 

Therefore,  ^=.10176^.  (194) 

Examples.  —  Let  Wf  =  6535  pounds. 
"     D  =  5feet. 

We  have  F=  .10176  -  -  133.05  square  inches. 


Again,         Let  Wf  =  1634  pounds. 
"     D  =  10  feet. 

We  have  F=  .10176  -  —  =  16.63  square  inches. 

(64.)  Balancing  the  Fly-Wheel.—  When  the  fly-wheel 
is  erected,  great  care  should  be  taken  that  its  centre  of  gravity 
coincides  with  the  centre  of  the  shaft  upon  which  it  is  placed. 

It  is  not  necessary  that  it  be  perfectly  circular,  so  long  as 
its  centre  of  gravity  coincides  with  its  axis,  as  will  presently 
be  shown. 

With  regard  to  a  plane  passing  through  its  centre  of 
gravity  at  right  angles  to  the  axis,  the  wheel  must  be  per- 
fectly symmetrical.  If  it  is  not,  the  centrifugal  force  will 
give  rise  to  a  couple  tending  to  place  it  in  that  position,  and 
of  course  straining  the  shaft  and  the  fly-wheel  unnecessarily. 
In  both  cases  these  disturbing  forces  will  increase  with  the 
square  of  the  velocity. 

If  any  two  masses,  as  W  and  w,  are  in  statical  equilibrium 
with  regard  to  an  axis  C;  if  caused  to  revolve,  they  will  have 
no  tendency  to  leave  their  common  axis. 


OF  THE  STEAM-ENGINE.  141 

FIG.  27. 


C 


Let  W=  weight  of  the  larger  mass. 
"     w  =  weight  of  the  smaller  mass. 
"     R  —  distance  of  the  larger  mass  W  from  C. 
"      r  —  distance  of  the  smaller  mass  w  from  C. 

Since  these  bodies  are  assumed  in  statical  equilibrium,  we 
have 


wv* 

The  centrifugal  force  of  the  smaller  body  =  -  . 

gr 

The  centrifugal  force  of  the  larger  body  = 


gR 

Letting  N  represent  the  number  of  turns  of  the  connected 
bodies  about  C  per  second,  we  have 

V=2*RN,  v 

TFF2 


and        -  «*  -  ,    and 

gr  gr  gR  gR 

And  dividing,  we  have 

wp 

gr         wr 
WVZ     WR' 
gR 

But  wr  =  WR.  Hence,  the  centrifugal  forces  of  the  two 
masses  are  equal  and  opposite,  which  was  to  be  shown. 

The  fly-wheel  is  sometimes  purposely  erected  out  of  bal- 
ance, in  order  to  force  the  crank  over  some  particular  point, 


142  THE  RELATIVE  PROPORTIONS 

but  this  method  of  proceeding  results  in  strains  upon  the 
shaft  and  its  bearings,  which  tend  to  injure  them  by  causing 
wear  and  vibration,  and  should  not  be  used  when  avoidable. 
Eefer  to  Art.  (35). 

(65.)  Speed  of  the  Rim  of  a  Fly-  Wheel.—  It  has  been 
stated  in  article  (61)  that  increasing  the  speed  of  the  rim  of 
a  fly-wheel,  or,  what  is  the  same  thing,  increasing  its  diam- 
eter and  number  of  revolutions,  one  or  both,  is  productive  of 
economy  in  its  weight. 

There  is,  however,  a  limit  beyond  which  the  speed  of  rim 
cannot  be  driven  without  bursting  the  rim. 

If  we  suppose  the  rim  to  be  solid,  and  neglect  the  sup- 
port that  it  receives  from  its  arms,  we  first  have  the  case  in 
which  centrifugal  force  acts  in  a  similar  manner  to  the  out- 
ward bursting-pressure  of  water.  (Weisbach's  Mechanics 
of  Engineering,  sec.  vi.,  art.  363.) 

Letting  T=ihe  tensile  strength  per  foot  of  area, 
"        —  =  mean  radius  of  rim  in  feet, 

"         u  =  velocity  of  rim  in  feet  per  second, 

G  =  weight  per  cubic  foot, 

"        p  =  radial  force  of  each  cubic  foot  due  to  centrif- 
ugal force, 

we    have   for   the  centrifugal  force  of   each    cubic    foot 


=  p  =  --  ,  and  also,  as  shown  in  Weisbach's  Mechanics  of 

gD 

2T 
Engineering,  the  resisting  force  is  p  =  -  .     Equating  these 

values,  we  have  T=  -  . 

9 


Therefore,  "''  (195) 


OF  THE  STEAM-ENGINE.  143 

Examples. — For  a  cast-iron  rim,  assuming  T=  2592000 
pounds  per  square  foot,  and  its  weight  G  =  450  pounds  per 
cubic  foot,  we  have  for  its  bursting  speed 


32.2x2592000     ,Qn-,    .  , 

u  =  -*/  —   — =  430.7  feet  per  second. 

If  we  use  a  factor  of  safety  of  10,  we  have 


132.2x259200     10CO-  , 

u  =  -*/ =  136.2  feet  per  second 

for  a  safe  speed. 

The  speed  of  rim  of  fly-wheels  is  in  some  cases  pushed  to 
about  80  feet  per  second,  but  is  probably  not  often  exceeded* 

It  is  of  interest  to  note  that  if  5000  pounds  per  square 
inch  be  regarded  as  a  safe  strain  for  a  railway  tire  when 
subjected  to  shocks  occurring  while  in  motion,  the  greatest 
speed  of  a  locomotive  with  safety  may  be  deduced. 


We  have     _     . . 

490 


u  =  */— -  —  =  217.|-  feet  per  second, 

•        4yu 

or  about  2.47  miles  per  minute. 


CHAPTER    XIV. 

(66.)  Centrifugal  Stress  on  the  Arms  of  a  Fly- 
Wheel. — In  the  case  just  discussed  the  fly-wheel  has  been 
supposed  to  rely  entirely  upon  the  strength  of  its  solid  rim, 
regardless  of  any  support  it  might  have  from  the  arms; 
when,  however,  large  wheels  are  constructed,  the  difficulties 
attendant  upon  handling  them,  as  well  as  the  expense  of 
making  large  castings,  make  it  necessary  to  cast  the  rim  in 
sections,  which  are  put  together  in  place,  the  arm  sometimes 


144 


THE  RELATIVE  PROPORTIONS 


FIG.  28. 


being  cast  with  its  section  of  rim,  which  is  the  best  method 
when  practicable,  and  sometimes  in  a  separate  piece.  These 
segments  on  being  put  in  place  are 
caused  to  abut  firmly  upon  each  other, 
and  held  in  position  by  means  of  a 
prisoner  and  keys,  as  shown  at  B,  or 
by  means  of  links,  as  shown  at  A, 
Fig.  28.  A  double-headed  bolt  some- 
times takes  the  place  of  the  link.  Both 
link  and  bolt  when  used  are  heated 
and  then  allowed  to  contract  in  place, 
drawing  the  ends  of  the  segments 
solidly  together. 

For  various  practical  mechanical 
reasons,  the  strength  of  the  bolts, 
prisoners  or  links  cannot  be  relied 
upon,  and  they  should  only  be  considered  as  fastenings  to 
preserve  the  form  of  the  fly-wheel. 

The  arm  for  each  segment  should  be  so  proportioned  as 
to  support  with  perfect  safety  its  weight  when  at  its  lowest 
position,  and  also  the  stress  due  to  its  centrifugal  and  tan- 
gential force,  a  slight  taper  being  given  to  the  arm,  increas- 
ing from  the  rim  to  the  hub,  to  allow  for  the  weight  and 
centrifugal  force  of  the  arm  itself. 

The  centrifugal  force  of  any  mass  is  exactly  what  would 
result  if  its  whole  mass  is  supposed  concentrated  at  its  cen- 
tre of  gravity  and  the  motion  of  this  point  alone  considered. 
We  have  (Weisbach's  Mechanics  of  Engineering,  sec.  iii., 
art.  115),  for  the  distance  from  the  centre  of  the  wheel  to 
the  centre  of  gravity  of  the  segment,  =  x, 


OF  THE  STEAM-ENGINE. 


145 


in  which  e  is  the  thickness  of  rim,  a  =  the  angle  A  C  B,  and 
D  is  its  mean  diameter.     Fig.  29. 


FIG.  29. 


Let  Wf  —  the  weight  of  the  rim  in  pounds. 
"     M  =  the  number  of  sections  =  number  of  arms  into 

which  the  rim  is  divided. 
"      (7= the  centrifugal  force  of  each  segment. 
"      u  =  the  velocity  of  the  rim  in  feet  per  second. 
"       v  -  the  velocity  of  the  centre  of  gravity  of  each 

segment. 


We  have 
and  since 
we  have 


M     gx 
u:v::D:2x, 
4zV 


Substituting  the  value  of  x  given  above,  we  have 


4  rr/      " 
Jf     ^D 


13 


146  THE  RELATIVE  PROPORTIONS 

In  this  latter  formula,  if  e  is  very  small  in  comparison  to 
jD,  we  can  neglect  the  small  term  ^  (  —  J  ,  and  the  equation 

i  n     A  Wf     u?  sin  .j 

becomes  0  =  4-—  -  —  . 

M        gDa 

If  to  this  expression  for  the  intensity  of  the  centrifugal 
force  of  each  segment  we  add  its  weight  for  its  lowest  posi- 
tion, we  have,  calling  ythe  strain  on  any  arm  in  the  direc- 
tion of  the  radius  of  the  wheel, 


(196) 


and  sioce  w2  -  .0006853924  D*N\  we  have 
.  00274156  D 


sinrl        ( 


ga 

in  which  g  =  32.2  feet  per  second. 

If  we  divide  the  intensity  of  the  force  Y  by  the  safe  strain 
in  tension  per  square  inch  of  the  material  of  the  arm,  we 
obtain  the  required  cross-section  to  resist  rupture  from  the 
centrifugal  force  and  weight  of  each  segment. 

Example.  —  Let  F^=1634  pounds.  Let  M=6  —  i.e.,  the 
rim  be  divided  into  six  parts,  each  having  its  arm.  Let 
D  =  10  feet  and  JV-  100  strokes  per  minute. 

We  also  have  -  -  30°,  and  a  -  y  =  1.0472. 


1388  pounds  approximately. 


OF  THE  STEAM-ENGINE.  147 

(67.)  Tangential  Stress  on  the  Arms  of  a  Fly« 
Wheel  for  a  Single  Crank. — In  addition  to  the  stress  on 
the  arm  due  to  its  weight  and  centrifugal  force,  each  arm 
sustains  a  tangential  strain  at  its  extremity  due  to  the  inertia 
of  the  rim,  which,  in  case  of  sudden  stoppages,  is  sometimes 
of  very  great  intensity. 

For  the  case  of  any  ordinary  fly-wheel,  whose  only  office  is 
to  equalize  the  work  given  out  in  each  element  of  time  or  arc, 

Letting  R  =  the  mean  radius  of  the  rim  in  feet, 

"      JT=the  sum  of  the  tangential  forces  at  the  extrem- 
ities of  the  arms  in  pounds, 

we  have,  for  the  work  given   out  or   absorbed   for  each 
element  da  of  arc, 

XRda. 

We  also  have,  equation  (33),  for  the  work  received  from 
the  steam-cylinder  for  each  element  da  of  arc, 
Sr  sin  a  da, 

and,  equation  (35),  for  the  work  given  out  uniformly  for  each 

2 
element  da  of  arc,  Sr-da. 

it 

We  see  that  the  rim  is  called  upon  at  certain  points  or 
during  certain  arcs  to  assist  or  resist  the  work  given  out  by 
the  steam-cylinder. 

If  we  place 

XRda  =  Sr  sin  a  da  -  Sr-da,  (198) 

7T 

v    ^ 
or  X 

we  obtain  the  measure  of  the  tangential  force  X. 
39°  32'  25" 


320°  27'  35" 


148  TEE  RELATIVE  PROPORTIONS 

.  For  a  =  90°  =  270°  we  have 

X=  +.36338—. 
For  a  =  0°  =  180°  we  have 

X= -.636618—.        (199) 

This  last  value  of  X  represents  the  maximum  value  of  X, 
being  its  value  for  the  two  dead  points ;  and  if  we  divide 
this  by  the  number  of  arms,  we  obtain  the  force  at  the  ex- 
tremity of  each  arm  which  tends  to  bend  or  to  break  it  at 
its  junction  with  the  hub  of  the  wheel,  or  the  rim.  We  have 
(Weisbach's  Mechanics  of  Engineering,  sec.  iv.,  art.  272)  the 
following  equation  for  a  beam  fixed  at  one  end  and  loaded 
at  the  other: 


'-ZJ*\   * 
F  \M)'  w 


w 

in  which  M  =  the  number  of  arms ;  F=  cross-section  of  an  arm 
in  square  inches ;  W=  its  measure  of  the  moment  of  flexure ; 
/  =  the  length  of  arm  in  inches ;  T=  the  proof  (or  safe)  stress 
per  square  inch ;  Y=  the  radial  stress  on  each  arm  ;  X=  the 
tangential  stress  on  each  arm  ;  and  e  =  the  half  diameter  of 
the  arm  in  the  plane  of  the  fly-wheel;  or  inversely, 

H{-£(i)4      «« 

In  this  formula,  for  round  and  elliptical  arms 

Fe     4 

~W~e' 

For  rectangular  arms, 

~W  =  e' 

(Weisbach's  Mechanics  of  Engineering,  Art.  236,  Sec.  iv.) 
In  formula  (200)  the  value  of  e  remains  to  be  determined 


OF  THE  STEAM-ENGINE.  149 

approximately.     This  can  be  done  by  substituting  in  either 
of  the  two  following  formulae,  and  taking  the  greater  value, 

:,•':•  -*!?-.  (201) 

(Weisbach's  Mechanics  of  Engineering,  Art.  235,  Sec.  iv.) 

•F-Jr-  (202) 

Example. — Let  us  assume  the  shape  of  the  arm  of  the  fly- 
wheel already  discussed  to  be  elliptical. 

X 

Y=  282  pounds,      —  =  141  pounds. 
M 

We  have,  equation  (202), 


Therefore, 

in  which  b  =  the  smaller  -^-diameter  of  the  arm  assumed, 

and,  equation  (201),  since  TF= for  an  ellipse, 

4 


(Weisbach's  Mechanics  of  Engineering,  Art.  231,  Sec.  iv.) 

Let  6-1",     T=1800,    and  J  =  60  inches. 

282x7 


/ 

V 


141x60x7 
22x1800 

13* 


150  THE  RELATIVE  PROPORTIONS 

The  second  value  of  e  =  2.44  inches  must  be  substituted  in 
formula  (200),  and  we  have 

If       4/X\  1  4x141x60 


=  7.862  square  inches. 
We  have  assumed  6  =  1  inch  ;  and  since  F=  neb,  we  have 

e  =  —  x  7.86  =  2.5  inches. 

We  thus  see  that  each  arm  of  a  fly-wheel  of  the  dimen- 
sions indicated  should  be  of  an  elliptical  form,  whose  major 
and  minor  axes  are  respectively  5  and  2  inches.* 

It  is  customary  to  give  the  arms  a  slight  taper  from  the 
hub  to  the  rim. 

(68.)  Work  Stored  in  the  Arms  of  the  Fly-  Wheel.— 
If  we  wish  to  take  into  account  the  weight  of  the  arms  in 
estimating  the  work  stored  in  the  fly-wheel,  we  have,  let- 
ting u  =  velocity  of  rim,  TPa  =  the  total  weight  of  the  arms, 
and  w  =  work  stored, 

W 

w  =  0.325  —  -u2,  approximately, 

which  can  be  added  to  the  work  stored  in  the  rim. 

For  a  more  general  and  less  practical  analytical  discus- 
sion of  fly-wheels,  reference  may  be  made  to  the  works  of 
Morin,  Dulos,  Poncelet  and  Resal. 

Dr.  R.  Proel,  in  his  Versueh  einer  Graphischen  Dynamik, 
gives  very  clear  and  elegant  graphical  methods  of  represent- 

*  If  the  power  of  the  engine  is  conveyed  by  means  of  a  band  or 
geared  fly-wheel,  we  must  calculate  the  tangential  stress  upon  the 
arms  by  means  of  the  theorem  of  moments,  regarding  the  crank  as 
the  short  lever  at  whose  extremity  the  whole  steam-pressure  acts. 


OF  THE  STEAM-ENGINE.  151 

ing  the  work  lost  and  gained  by  a  fly-wheel  under  various 
conditions. 

It  perhaps  appears  superfluous  to  some  of  our  readers  to 
enter  into  detail  to  so  great  an  extent  as  has  here  been  done, 
but  the  danger  and  loss  resulting  from  the  accidental  breakage 
of  a  fly-wheel  demand  the  most  painstaking  care  in  estab- 
lishing its  dimensions. 

(69.)  The  Working -Beam. — The  working-beam  is  be- 
coming less  used  as  the  speed  of  the  steam-engine  is  increased ; 
it  is  preferably  constructed  of  wrought  iron  or  steel,  or,  if 
made  of  cast  iron,  is  in  many  instances  bound  around  with 
wrought  iron.  Its  form,  if  solid,  should  be  parabolic,  with 
the  vertex  at  the  point  where  the  connecting-rod  joins  it, 
and  the  load  at  that  point  is  the  total  pressure  of  the 
steam  upon  the  piston-head.  (Weisbach's  Mechanics  of  En- 
gineering, sec.  iv.,  arts.  251-52-53.) 

The  working-beam  is  supposed  to  be  fixed  at  its  central 
bearing,  and  thus  becomes  a  beam  fixed  at  one  end  and 
loaded  at  the  other.  See  Table  VI. 

Where  web-bracing  is  used  in  working-beams,  the  graphi- 
cal method  will  afford  the  simplest  solution.  (See  Graphical 
Statics,  Du  Bois.) 

(70.)  General  Considerations. — The  recent  improve- 
ments in  parallel  motions  will  probably  lead  to  their  more 
general  use  in  the  place  of  guides  and  slides.  A  most  in- 
teresting and  instructive  little  work,  How  to  Draw  a  Straight 
Line,  by  A.  B.  Kempe,  suggests  to  the  mechanician  many 
forms  which  can  be  adapted  to  the  steam-engine  with  little 
trouble. 

In  the  foundation  and  framework  of  engines  every  pre- 
caution must  be  taken  to  obtain  RIGIDITY  and  immova- 
bility. 

Too  much  stress  cannot  be  laid  upon  this  point ;  an  in- 


152  THE  RELATIVE  PROPORTIONS 

secure  foundation  inevitably  injures,  and  perhaps  ruins,  the 
engine  upon  it. 

The  centrifugal  governor  has  not  been  considered,  because 
it  forms  one  of  the  principal  topics  in  almost  every  work  on 
the  steam-engine. 

The  practical  defects  of  the  centrifugal  governor  are  in- 
surmountable when  a  perfectly  regular  speed  is  desired  of 
the  engine.  They  are  as  follows : 

(1.)  The  engine  must  go  fast  in  order  to  go  slow,  or  the 
reverse,  since  the  balls  cannot  move  without  a  change  of 
speed  in  the  engine  and  themselves. 

(2.)  The  opening  of  the  steam-valve  is  dependent  upon 
the  angle  which  the  arms  attached  to  the  balls  form  with 
the  central  spindle  around  which  they  revolve. 

Thus,  an  engine  having  its  full  amount  of  work,  and  gov- 
erned by  an  ordinary  ball-governor,  will  be  kept  at  a  uni- 
form speed  by  the  governor  so  long  as  the  average  resistance 
to  be  overcome  by  the  engine  remains  constant ;  but  when- 
ever any  of  the  work  is  taken  off,  the  speed  of  the  engine 
will  be  increased  to  a  higher  rate,  corresponding  to  the 
diminished  work,  and  at  this  faster  speed  the  engine  will 
then  run  uniformly  under  the  mastery  of  the  governor  so 
long  as  the  work  continues  without  further  alteration.  This 
arises  from  the  fact  that  the  degree  of  opening  of  the  steam- 
valve  is  directly  controlled  by  the  angle  to  which  the  gov- 
ernor-balls are  raised  by  their  velocity  of  revolution,  the 
steam-valve  being  moved  only  by  a  change  of  speed,  and 
consequently  by  a  change  of  the  angle  of  suspension  of  the 
governor-balls ;  whence  it  follows  that  a  larger  supply  of 
steam  for  overcoming  any  increase  of  work  can  be  obtained 
only  in  conjunction  with  a  smaller  angle  of  the  suspension- 
rods  of  the  governor-balls,  and  consequently  with  a  slower 
speed,  and  that  a  larger  angle  of  the  ball-rods,  and  con- 
sequently a  higher  speed,  must  be  attained  in  order  to  reduce 


OF  THE  STEAM-ENGINE.  153 

the  supply  of  steam  for  meeting  any  reduction  of  work  to 
be  done  by  the  engine. 

(3.)  The  governor  must  be  sensitive — i.  e.,  quick  to  act. 
This  result  is  usually  attained  in  the  centrifugal  governor  by 
giving  to  the  balls  a  speed  much  greater  than  that  of  the 
engine,  so  that  a  slight  variation  of  speed  in  the  engine  is 
multiplied  in  the  governor  many  times. 

A  high  speed,  however,  is  attended  with  the  disadvantage 
of  rapid  wear,  and,  in  the  case  of  an  ordinary  governor, 
wear  such  as  to  admit  of  any  lost  motion  is  attended  with 
much  trouble  to  the  engineer  and  sudden  variations  of  speed 
in  the  engine. 

(4.)  The  governor  must  have  power,  which  means  an 
even  and  sure  motion  of  the  valve  notwithstanding  the 
almost  unavoidable  defects  of  workmanship,  such  as  the 
sticking  of  the  valve  or  the  binding  of  the  valve-stem 
through  careless  packing  of  the  stuffing-box.  In  the  ordi- 
nary governor  this  power  is  sought  to  be  obtained  either  by 
a  high  speed,  the  defects  of  which  have  already  been  pointed 
out,  or  by  means  of  very  heavy  balls,  which  results  in  a  very 
cumbersome  and  large  machine,  besides  adding  largely  to 
the  expense. 

Thus  we  see  that  not  only  is  the  speed  of  the  steam-engine 
entirely  different  with  different  loads,  but  also  that  with  a 
constant  load  the  speed  varies  between  limits  which  are  de- 
termined by  the  sensitiveness  of  the  governor  and  is  at  no 
time  regular. 

The  necessity  of  a  very  sensitive  governor  is  done  away 
with  by  the  use  of  a  properly  proportioned  fly-wheel. 

The  use  of  the  governor  to  determine  the  point  of  cut-off, 
as  shown  in  the  Corliss  engines,  if  the  fly-wheel  be  of  the 
proper  weight  and  size,  is  attended  with  great  regularity  of 
speed  and  economy  of  steam.  Siemens'  chronometric  gov- 
ernor, in  which  first  the  inertia  of  a  pendulum  and  after- 


154  THE  RELATIVE  PROPORTIONS 

ward  hydraulic  resistance  were  used  as  a  point  d'appui  to 
move  the  valve  from,  produces  a  very  regular  speed  of  en- 
gine (Proceedings  of  the  Institution  of  Mechanical  Engineers, 
January,  1866),  but  is  too  costly  for  general  use. 

Marks'  isochronous  governor  (patented),  in  which  the 
motion  of  the  valve  precedes  any  change  of  speed  in  the 
governor-balls,  or.  as  since  altered,  in  the  hydraulic  cup, 
subserves  the  same  purpose,  and  is  much  cheaper  than  the 
former.  (Journal  of  the  Franklin  Institute,  May,  1876.) 

A  vast  number  of  forms  of  governor  of  varying  merit 
have  been  invented,  this  portion  of  the  steam-engine  appear- 
ing to  be  the  most  attractive  to,  and  the  most  considered  by, 
mechanics  and  engineers. 

The  only  test  of  beauty  and  elegance  of  design  in  an  en- 
gine is  fitness  and  perfect  proportion  to  the  stresses  placed 
upon  the  various  parts. 

The  severest  simplicity  of  design  should  be  adhered  to. 
Every  pound  of  metal,  where  it  does  not  subserve  some  use- 
ful purpose,  every  attempt  at  mere  ornament,  is  a  defect,  and 
should  be  avoided. 

The  well-educated  engineer  should  combine  the  qualities 
of  the  practical  man  and  the  physicist ;  and  the  more  he 
blends  these  together,  making  each  mould  and  soften  what 
the  other  would  seem  to  dictate  if  allowed  to  act  alone,  the 
more  will  his  works  be  successful  and  attain  the  exact  object 
for  which  they  are  designed. 

The  steam-boiler  and  its  construction  will  be  found  to  be 
very  thoroughly  treated  in  Professor  Trowbridge's  Heat  and 
Heat  Engines,  in  Wilson's  Steam-boilers,  and  in  The  Steam- 
Engine,  by  Professor  Rankine. 

(71.)  Note  on  the  Taper  of  Connecting-Rods.— The 

following  remark,  page  62,  needs  qualification  : 

"  It  is  customary  to  make  round  connecting-rods  with  a 


OF  THE  STEAM-ENGINE.  155 

taper  of  about  one-eighth  of  an  inch  per  foot,  from  the 
centre  to  the  necks,  which  should  be  of  the  calculated  di- 
ameter. Experiment  does  not  show  an  increased  strength 
from  a  tapering  form." 

Passed  Assistant  Engineer  C.  H.  Manning,*  in  a  pleasant 
correspondence  with  the  writer,  differed  from  him,  deeming 
it  a  better  method  to  taper  connecting-rods  from  the  crank- 
pin  end  to  the  cross-head  end,  as  "  experience  had  shown 
that  connecting-rods  usually  failed  at  the  crank-pin." 

Led  by  this  discussion  to  make  a  more  thorough  inves- 
tigation into  the  stress  upon  connecting-rods  due  to  their 
own  inertia  than  he  had  before  deemed  necessary,  the  writer 
submits  the  results,  hoping  they  may  be  of  interest  to  engi- 
neers engaged  in  the  designing  of  mechanism. 

The  connecting-rod,  if  of  wrought  iron,  must  be  considered 
either,  1st,  as  a  short  column,  tending  to  rupture  by  crushing, 
or  2d,  as  a  long  column,  tending  to  rupture  by  buckling.  The 
tendency  to  fail  in  tension  can  be  neglected,  as  being  much 
less  than  in  the  two  cases  mentioned. 

In  the  first  case  it  is  obvious  that  tapering  will  not  add 
to  its  strength  if  we  neglect  the  stress  due  to  its  inertia  and 
weight. 

In  the  second  case,  theoretically,  if  we  disregard  the  stress 
in  flexure  due  to  the  inertia  and  weight  of  the  connecting- 
rod,  the  increase  in  its  diameter  will  be  a  maximum  at  its 
centre.  (See  Weisbach's  Mechanics  of  Engineering,  Sec.  iv., 
Art.  267.) 

The  connecting-rod,  if  of  steel,  may  be  considered,  1st,  as 
a  tension-rod,  tending  to  fail  in  tension,  or  2d,  as  a  long 
column,  tending  to  rupture  by  buckling.  The  ability  of 
steel  to  withstand  a  much  greater  stress  in  compression  than 
in  tension  avoids  the  necessity  of  considering  it  as  a  short 

*  Instructor  in  steam  engineering,  U.  S.  Naval  Academy,  Annap- 
olis, Md. 


156 


THE  EELATIVE  PROPORTIONS 


column,  failing  by  crushing.     In  both  cases  the  inertia  and 
weight  of  the  rod  are  disregarded. 

These  are  the  general  conditions  which  have  controlled 
the  mechanical  engineer  in  the  consideration  of  the  propor- 
tions of  wr ought-iron  or  steel  connecting-rods.  In  addition, 
it  has  been  customary,  in  order  to  meet  an  unknown  stress 
in  flexure  due  to  the  inertia  of  the  rod,  to  increase  the 
diameter  of  the  rod  at  the  middle,  or  latterly  to  give  an 
increased  diameter  at  the  crank-pin,  and  taper  from  this  to 
the  smallest  dimension  of  the  rod  at  the  necks  or  neck,  this 
increase  being  purely  empirical. 

FIG.  30. 


Another  correspondent  says :  "  Where  iron  and  steel  are 
used,  the  figures  adopted  as  constants  must  vary  greatly, 
especially  with  steel.  The  value  of  E  ( modulus  of  elas- 
ticity) varies  enormously;"  with  which  opinion  the  writer 
agrees  perfectly,  only  regretting  that  the  very  basis  of 
all  correct  calculations  is  thus  taken  from  us,  and  we  are 
forced  to  be  contented  with  results  which  have  a  reasonable 
probability  of  correctness  if  we  assume  average  values  for 
the  safe  stresses  per  square  inch  and  the  moduli  of  elasticity 
of  the  material  with  which  we  are  dealing. 

If  a  straight  line,  A  B,  Fig.  30,  have  its  extremities  A 
and  B  respectively  caused  to  move  reciprocally  upon  the 


OF  THE  STEAM-ENGINE.  157 

straight  line  EAH  and  in  the  perimeter  of  the  circle 
BFG,  any  point  upon  the  line  AB  as  d  will  trace  an 
approximate  ellipse  as  d  H  e  F. 

If,  now,  we  let  the  length  of  the  line  AB  =  l;  the  radius 
CB  of  the  circle  =  r ;  the  variable  distance  of  the  point  d 
from  the  point  A  =  Ad  =  x ;  we  have  the  semi-major  axis 
of  this  ellipse  hF=hH=r,  and  the  semi-minor  axis  hd  = 

y  ' 

he  =  -r,  and  the  radius  of  curvature  of  the  osculatory  circle 

I 

at  the  vertex  d  of  the  semi-minor  axis, 

R  =  -r.  (203) 

x 

The  maximum  resistance  due  to  its  inertia  of  any  small 
mass  m  at  the  end  B  of  the  rod  A  B,  to  motion  in  the  direc- 
tion B  C,  is  equal  to  its  centrifugal  force, 

mv2 


(204) 


in  which  v  =  the  linear  velocity  of  the  point  B  in  feet  per 
second. 

(For  a  demonstration  of  this  refer  to  a  paper  by  F.  A.  P. 
Barnard,  Transactions  of  the  American  Institute.) 

Observing  now  that  in  the  position  shown  in  the  figure 
every  point  on  the  line  A  B  as  d  is  moving  with  a  velocity 

v  in  an  arc  of  an  osculatory  circle  whose  radius  is  R  =  -r, 

x 

equation  (203),  we  have  the  means  of  determining  the  re- 
sistance due  to  the  inertia  of  each  element  of  mass  m. 
For  the  point  d  it  would  be 

IT-IT  (205) 

for  the  point  A,  since  x  =  0,  R  =  «,  and  the  resistance  of  a 

14 


158 


THE  RELATIVE  PROPORTIONS 


mass  w  would  be  =  0.  We  thus  see,  equation  (205),  that 
the  resistance  of  each  element  of  mass  due  its  inertia  to 
motion  in  a  direction  at  right  angles  to  E  A  C  is  directly 
proportional  to  its  distance  =x  from  the  point  A. 

If,  now,  for  the  line  A  B  we  substitute  a  rod  of  uniform 
cross-section  =F,  we  have 


Frdx 


(206) 


in  which  Y  =  weight  per  cubic  unit,  and  #  =  32.2  feet  per 
second,  F  being  taken  in  square  units. 

Substituting  this  value  of  m  in  equation  (205),  and  in- 
tegrating between  the  limits  x  =  0  and  =  £,  we  have,  denoting 
the  whole  resistance  of  the  rod  A  B  by  P, 


glr 


xdx 


or  since  the  weight  of  the  rod  —  G  =  Ffl, 


(207) 


And  since  this  load  upon  the  rod  due  to  the  resistance  of  its 
own  inertia  increases  uniformly  from  the  end  A  to  the  end 
B,  the  rod  can,  with  sufficient  approximation,  be  supposed 


FIG.  31. 


L9 


OF  THE  STEAM-ENGINE.  159 

in  the  condition  of  a  horizontal  beam  loaded  with  a  tri- 
angularly-shaped load,  as  shown,  Fig.  31,  in  the  cross- 
hatched  portion  A  KB. 

For  a  vertical  engine  we  can  neglect  the  weight  of  the 
connecting-rod,  considering  only  the  stress  due  to  its  inertia. 

For  the  moment  of  flexure  of  any  cross-section  at  a  dis- 
tance x  from  the  extremity  A  we  have,  letting  c  =  the  load 
per  unit  of  length  at  the  point  Kt 

(208) 

which  becomes  a  maximum  for  #  =  £l/|f  =  0.578/,  at  which 
point,  therefore,  the  maximum  cross-section  of  the  connect- 
ing-rod should  be  placed. 

In  horizontal  engines  it  is  necessary  to  take  into  consid- 
eration the  weight  of  the  rod,  as  well  as  its  inertia ;  the 
weight  of  the  rod  may  be  regarded  as  a  uniformly  dis- 
tributed load  always  acting  in  one  direction. 

For  the  cross-section,  at  a  distance  x  from  the  extremity 
A,  the  moment  of  flexure  is,  letting  G  =  whole  weight  of 
rod  =  rl, 


"lf~Wf 

which  is  a  maximum  for 


But  we  have,  from  equation  (207), 

1       fir 
=  *p  in  which  g  =  acceleration  of 

velocity  of  the  crank-pin,  J5,  in  feet  per  second,  and  r  =  the 


C1       fir 
-  =  *p  in  which  g  =  acceleration  of  gravity,  v  =  the  linear 


160  THE  RELATIVE  PROPORTIONS 

radius  of  the  crank  in  feet.   Substituting  this  value  in  equa- 
tion (210),  we  have 


which  shows  that  as  the  velocity  of  the  crank-pin  increases 
the  value  of  x  approaches  more  nearly  to  /  V\  =  0.578Z,  but 
can  never  quite  equal  it. 

Referring  to  equation  (209),  and  taking  P  =  zero,  we  find 

the  maximum  value  of  x  =  —  0.5£ 

2i 

We  thus  see  that  the  greatest  moment  of  flexure  of  any 
connecting-rod  lies  between  the  limits  0.5/  and  0.578£,  mea- 
sured from  the  point  A. 

It  is  of  interest  further  to  note  that  the  crank-pin  takes 
one-half  the  stress  due  to  the  weight  of  the  rod,  and  two- 
thirds  of  the  stress  due  to  the  inertia  of  the  rod. 

The  end,  A,  takes  one-third  of  the  stress  due  to  the 
inertia  of  the  rpd,  alternately  increasing  and  decreasing 
the  stress  upon  the  guides  to  this  amount,  and  one-half  the 
stress  due  to  its  weight. 

In  engines  having  a  large  number  of  revolutions  per 
minute  P  becomes  worthy  of  notice.  In  slow-moving  en- 
gines it  is  very  small,  and  may  be  neglected. 

From  these  considerations  the  writer  is  of  the  opinion 
that  the  failure  of  connecting-rods  at  the  neck  nearest  the 
crank-pin,  if  they  are  properly  proportioned,  is  probably  due 
to  the  crank-pins  being  out  of  truth,  or  to  the  seizing  of 
heated  boxes  on  the  crank-pin,  rather  than  to  the  weight 
or  inertia  of  the  rods  themselves. 

The  assumption  of  a  connecting-rod  of  uniform  cross- 
section  is  manifestly  incorrect,  and  only  serves  to  prove 
generally  a  principle  which  should  be  recognized. 

In  the  case  of  a  practical  application,  graphical  methods 


OF  THE  STEAM-ENGINE.  161 

taking  cognizance  of  the  usual  changes  in  cross-section 
must  be  used  for  a  series  of  approximations.  (Vide  Du 
Bois's  Graphical  Statics,  or  Reuleaux,  Der  Constructeur.') 


CHAPTER    XV. 

(72)  The  Limitations  of  the  Steam-Engine.— Per- 
haps there  is  no  more  unsafe  proceeding  in  science  than  to 
attempt  to  predict  the  limitation  of  the  development  of  any 
of  its  results.  Yet  the  theory  of  the  steam-engine  has  so 
far  been  developed  as  to  seem  to  permit  us  to  mark  out  the 
boundaries  of  its  progress  with  tolerable  accuracy. 

At  least,  we  trust  that  a  full  criticism  of  its  present  de- 
ficiencies, and  a  deduction  of  its  limitations  from  known 
laws,  will  interest  such  of  our  readers  as  are  engaged  in 
the  improvement  of  this  machine. 

In  this  lecture  we  will  consider  the  steam-engine  alone, 
as  it  is  in  no  wise  responsible  for  the  economy  of  the  boiler 
which  supplies  the  steam,  or  for  the  losses  incurred  in  con- 
veying the  steam  from  the  boiler  to  the  engine. 

The  boiler  should  be  considered  apart  from  the  engine ; 
and  it  is  well  to  incidentally  remark  here  that  the  perform- 
ances of  the  best  types  of  boiler  leave  but  little  room  for 
improvement  so  far  as  evaporation  is  concerned. 

It  is  rather  in  the  utilization  of  the  steam  after  it  reaches 
the  engine  that  we  must  look  for  progress,  and  in  this  point, 
we  take  it,  lies  the  value  of  a  thorough  investigation  of  the 
machine  itself. 

The  well-known  formula  for  the  horse-power  of  a  steam- 
engine  *  contains  three  factors,  which  can  be  varied  at  will 

* =  (HP] :  in  which  P=  mean  effective  pressure  per  square 

33000 

14* 


162  THE  RELATIVE  PROPORTIONS 

— the  mean  effective  pressure,  the  volume  of  the  cylinder, 
and  the  number  of  strokes  per  minute. 

Concentration  of  power  in  a  small  space  is  the  greatest 
attribute  of  the  steam-engine ;  it  is  its  power  to  concentrate 
the  strength  of  thousands  of  horses  in  the  space  of  an  ordi- 
nary room  that  renders  the  steam-engine  so  useful  and  in- 
dispensable to  us. 

For  this  reason  we  cannot  go  on  increasing  the  volume 
of  the  steam-cylinder  indefinitely  without  a  proportionate 
increase  in  power ;  and  it  is  only  necessary  to  remark  to 
mathematicians,  with  regard  to  the  proportions  of  the  cylin- 
der, that  that  cylinder  whose  stroke  equals  its  diameter  con- 
tains the  maximum  of  volume  with  the  minimum  of  con- 
densation-surface.* 

The  importance  of  reducing  condensation-surface  is  be- 
coming more  and  more  appreciated  among  engineers,  and 
shortening  the  stroke  will  prove  less  of  a  bugbear  when  the 
laws  of  inertia,  as  applied  to  the  reciprocating  parts  of  a 
steam-engine,  are  better  understood. 

Shortening  the  stroke  has  the  further  advantages  of  short- 
ening the  space  passed  over  by  the  piston-head  during  a 
given  number  of  strokes,  and  consequently  of  reducing  the 
wear  on  the  piston-head,  as  well  as  of  rendering  the  engine 
more  compact  for  a  given  power. 

The  following  formula  gives  for  any  assumed  horse-power, 
mean  effective  pressure,  and  number  of  strokes  per  minute, 

inch ;  L  =  length  of  stroke  in  feet ;  A  =  area  of  piston  in  square  inches ; 
N=  number  of  strokes  per  minute;  (Jf?P)  =  indicated  horse-powers; 

— —  =  V=  volume  of  cylinder  in  cubic  feet. 
144 

*  Should  the  internal  condensation  of  cylinder-surface  prove  to  be 
directly  proportional  to  the  time  of  exposure,  as  well  as  to  the  area 
exposed,  the  stroke  should  be  twice  the  diameter;  but  the  writer 
doubts  the  practical  accuracy  of  this  statement  in  high-speed  engines. 


OF  THE  STEAM-ENGINE.  163 

the  required  common  diameter  and  stroke  of  cylinder,  in 
inches  : 

<*  =  79.59/'  ( 

Concentration  being  attained,  the  next  important  consid- 
eration that  arises  is  economy  of  steam  ;  and  this  is  the  battle- 
ground upon  which  the  struggle  for  supremacy  among  our 
engine-builders  is  to-day  being  practically  fought  out,  while 
the  physicists  are  quite  as  hotly  engaged  in  a  dispute  as  to 
the  relative  theoretical  merits  of  steam-,  hot  air-,  and  electric- 
engines,  into  which  we  will  not  enter. 

Practically,  we  think  that,  as  far  as  the  steam-engine  is 
concerned,  the  impossibility  of  supplying  to  the  steam  in 
the  cylinder  any  appreciable  amount  of  heat  by  means  of 
steam  or  hot-air  jackets  is  acknowledged  by  all,  and  the 
jacket  is  only  expected  to  keep  the  cylinder  warm  and  pre- 
vent it  from  abstracting  heat  from  the  steam  inside  the 
cylinder. 

The  jacket  proving  imperfect,  we  must  find  other  means 
for  preventing  condensation  by  cooling ;  and  they  are  these : 
Diminution  of  condensation-surface,  as  already  stated  above, 
and  shortening  of  the  duration  of  the  time  for  condensation, 
which  means  an  increase  in  the  number  of  strokes  of  the 
piston.  The  effect  upon  the  structure  of  the  engine  of  in- 
creased rotative  speed  is  not  injurious  to  the  engine,  as  so 
positively  stated  by  many  engineers ;  but,  if  attempted,  rapid 
speed  requires  a  far  greater  knowledge  of  the  dynamics  of 
the  steam-engine  than  has  as  yet  been  applied  to  it  by  many 
constructors,  and  will  almost  inevitably  shake  to  pieces  a 
faultily  designed  structure. 

The  length  of  the  crank-pin  and  shaft-bearings  must  also 
increase  directly  with  the  number  of  revolutions  made,  while 
their  diameter,  within  ordinary  limits  as  to  pressure  and 
speed,  has  no  appreciable  effect  upon  the  heating  of  the 


164  THE  RELATIVE  PROPORTIONS 

bearing ;  but  when  we  consider  the  practical  fact  that  the 
eccentrics,  possibly  because  of  greater  speed  of  rubbing  sur- 
faces, are  always  sources  of  large  frictional  losses,  it  would 
seem  best  to  make  the  diameters  of  all  bearing  journals  just 
as  small  as  the  pressures  allowable  upon  the  surfaces  of 
contact  will  permit.  Much  yet  remains  to  be  determined 
with  regard  to  the  laws  controlling  bearings. 

It  has  been  shown  conclusively  that  the  coefficient  of 
friction  is  much  affected  and  reduced  by  the  state  of  the 
rubbing  surfaces,  by  the  method  of  lubrication,  and  by  the 
quality  of  the  lubricant. 

Perfect,  almost  mirror-like,  surfaces  reduce  the  coefficient 
of  friction  far  below  the  3  to  5  per  cent,  formerly  stated  by 
Morin  as  a  minimum,  provided  a  continuous  lubricating 
apparatus  is  used  with  good  sperm  oil ;  and  it  will  be  found 
that  a  costly  lubricant,  such  as  sperm  oil,  is  cheapest  where 
care  is  exercised  not  to  waste  in  its  application.* 

The  important  influence  of  the  number  of  turns  upon  the 
length  of  bearings  has  not  been  clearly  enough  understood 
to  prevent  accidents  from  heating  of  the  crank-pins  and 
bearings,  which  is  one  of  the  most  annoying  mishaps  which 

*  Let          Z  =  the  length  of  the  crank-pin,  in  inches. 
"          L=  "  stroke,  in  feet. 

"    ( HP}  =  the  indicated  horse-power. 

Then,  for  marine  propeller  engines  a  practical  formula,  giving  safe 
lengths,  would  be 

^0.622 

One-fourth  of  this  length  will  serve  safely  for  side-wheel  or  stationary 
engines. 

For  locomotive  engines  practice  seems  to  prove  that  i  =  .013cP  is  a 
safe  value,  in  which  d  =  diameter  of  the  steam -cylinder,  in  inches. 

For  crank-shaft  bearings,  letting  R  =  the  whole  maximum  weight 
on  bearings,  and  N=  the  number  of  strokes  per  minute,  we  have 
/  =  . 000005  RN. 


OF  THE  STEAM-ENGINE.  165 

can  befall  an  engine,  and  at  times  requires  the  utmost  vig- 
ilance of  the  engineer  for  its  prevention. 

The  great  difficulty  attendant  upon  making  bearing  and 
shaft  press  uniformly  throughout  the  length  of  the  journal 
has  almost  forced  the  use  of  Babbitt  or  soft  metal  for  high- 
speed engines  ;  and  where  proper  care  is  taken  in  propor- 
tioning the  results  have  justified  its  use. 

Crank-pins  shrunk  in  by  heating  the  crank  are  apt  to  be 
out  of  truth,  no  matter  how  carefully  the  work  has  been 
done,  as  heat  is  apt  to  warp  the  crank. 

Pins  should  be  forced  in  cold,  or  the  crank  and  pin  be 
forged  in  one  piece,  as  in  the  case  of  most  small  double 
cranks. 

High  rotative  speed,  while  increasing  the  power  of  the 
steam-engine,  also  renders  it  more  compact  and  diminishes 
the  weight  of  the  fly-wheel  necessary  to  obtain  regularity 
of  speed. 

Rotative  speed  has,  however,  its  limitations,  independ- 
ently of  the  trouble  sometimes  caused  by  the  heating  of 
bearings. 

For  instance,  it  can  be  shown  that  if  the  safe  stress  in 
tension  upon  cast  iron  equals  1800  pounds  per  square  inch, 
a  cast-iron  ring  can  be  revolved  safely  at  a  maximum  lineal 
speed  of  about  8000  feet  per  minute,  which  at  once  places  a 
limitation  upon  the  rotative  speed  of  the  fly-wheel,  and 
demands  reduction  of  its  diameter  as  the  rotative  speed  is 
increased. 

Let  JV=the  number  of  revolutions  per  minute. 
"    D  =  the  diameter  of  cast-iron  ring,  in  feet. 
We  have 


or  .  DN-254G. 

This  formula  would  limit  the  diameter  of  the  fly-wheel, 
if  used,  of  a  pair  of  engines  to  make  600  revolutions  per 


166  THE  RELATIVE  PROPORTIONS 

minute,  lately  proposed  by  the  distinguished  engineer,  Mr. 
Charles  T.  Porter,  to  about  4  feet. 

Of  course,  as  the  tensile  strength  of  the  material  of 
which  the  fly-wheel  is  made  is  increased,  the  lineal  speed 
at  which  it  can  be  driven  is  increased. 

If  we  take  5000  pounds  per  square  inch  as  the  safe  ten- 
sile strength  of  the  steel  tire  of  a  locomotive,  we  find  its 
lineal  speed  to  be  limited  to  about  2^  miles  per  minute,  or 
150  miles  per  hour,  and  it  only  remains  for  our  engineers 
to  provide  boiler-capacity  and  road-bed  adapted  to  such 
speeds  in  order  that  they  may  be  realized  by  a  properly 
constructed  locomotive. 

This  speed  would  shorten  the  time  between  New  York 
and  Philadelphia  (90  miles)  to  36  minutes. 

High  rotative  speed  necessitates  special  precautions  as  to 
the  weight  of  the  reciprocating  parts. 

If  we  impose  the  condition  that  the  initial  and  final 
stresses  upon  the  crank-pin  be  equal,  we  must  make  the 
centrifugal  force  of  the  reciprocating  parts  (piston,  piston- 
rod,  cross-head,  and  connecting-rod),  supposed  to  be  concen- 
trated at  the  crank-pin,  equal  to  half  the  difference  of  the 
initial  and  final  steam-pressures  upon  the  whole  piston-head. 

High  rotative  speed  has  also  demanded  a  change  iii  the 
form  of  the  connecting-rod,  which  was  formerly  largest  at 
its  mid-length,  but  has  of  late  years  been  made  largest  at 
the  crank-pin  end ;  a  consideration  of  the  stress  upon  it  due 
to  its  own  inertia  would  place  the  point  of  greatest  stress 
at  0.578  of  its  length  from  the  cross-head  end;  but  the 
accidental  stresses  due  to  the  twisting  of  the  crank-pin  in 
its  boxes  recommends  as  a  measure  of  safety  that  the  rod 
be  made  largest  at  the  crank-pin  neck,  and  tapered  down 
from  that  to  the  cross-head. 

In  a  properly  proportioned  engine  of  any  assumed  horse- 
power the  only  method  of  reducing  the  weight  of  the  moving 


OF  THE  STEAM-ENGINE.  167 

parts  is  to  increase  the  number  of  strokes  per  minute ;  length- 
ening the  stroke  will  not  do  it. 

The  points  of  interest  with  regard  to  the  volume  of  the 
steam-cylinder  and  the  rotative  speed  of  the  engine  having 
been  touched  upon,  there  remains  to  be  considered  the  mean 
effective  pressure  of  the  steam. 

In  the  Journal  of  the  Franklin  Institute  for  June,  1880, 
the  writer  has  shown,  under  the  assumption,  approximately 
correct,  that  the  steam-expansion  curve  is  an  equilateral 
hyperbola,  that  the  point  of  cut-off  giving  greatest  econ- 
omy of  steam,  but  not  of  money,  for  steam-engines,  is 
determined  by  dividing  the  absolute  back  pressure  by  the 
absolute  initial  pressure  of  the  steam. 

Indicators  do  not  give  such  curves,  even  if  in  perfect 
order,  unless  the  steam  is  a  little  wet,  but  the  approxima- 
tion is  sufficiently  close  for  all  practical  purposes. 

A  more  accurate  way  would  be  to  say  that  the  most 
economical  point  of  cut-off  is  attained  when  the  final  steam- 
pressure  equals  the  back  pressure ;  but  our  first  statement  is 
more  convenient  in  form,  and  will  serve  our  purpose  with  all 
the  practical  accuracy  necessary. 

Let  P= the  mean  effective  pressure  of  steam  in  pounds 

per  square  inch. 
"    Pb  =  the  initial  absolute  pressure  of  steam  in  pounds 

per  square  inch. 
"    B  =  the  back  absolute  pressure  on  piston  in  pounds 

per  square  inch. 
"     e  =  the  fractional  part  of  the  stroke  at  which  the 

steam  is  cut  off. 

"    F=the  volume  of  the  steam-cylinder  in  cubic  feet. 
Then  we  have 

r  fi 

P=ePb\\  +nat.  log-  -B  (212) 


L1 

Let  (HP)  =  the  indicated  horse-power. 


•j 


168  THE  RELATIVE  PROPORTIONS 

Let  $  =  the  specific  volume  of  the  steam  for  a  pres- 
sure, Pb. 

"    TF"=the  weight  of  water  used  per  horse-power. 
We  have 


33000 ' 

in  which  L  =  the  length  of  stroke  in  feet. 

N  =  the  number  of  strokes  per  minute. 

A  =  the  area  of  the  piston-head  in  square  inches. 
Therefore 

f  -tl  -v 

VN 


log  - -B\ 

C     J 


(J?P)=- L J : XH4,   -(213) 

33000 

since  144  V=LA. 

The  quantity  of  water  used  during  one  minute  is 

>  (214> 

o 

62J  pounds  being  taken  as  the  weight  of  one  cubic  foot  of 
water,  and  S  representing  the  specific  volume  of  the  steam 
for  the  pressure  P6. 

Therefore,  the  weight  of  water  used  per  horse-power  is 


144 
33000  , 

143236 


S    ePb    l-mat  log     -  B    .  (215) 

j) 

If  in  this  last  formula  we  substitute  —  for  e,  we  obtain 


OF  THE  STEAM-ENGINE.  169 

the  theoretical  minimum  quantity  of  water  to  be  evaporated 
per  horse-power,  and 

W  __  14323         _      6220.3 


For  one-horse-power  per  hour  we  would  have,  multiply- 
ing by  60, 

60  W=     873218    .  (217) 

P»S  log 


As,  for  instance,  for  a  non-condensing  engine  having  a 
gauge-pressure  of  45  pounds,  and  cutting  off  at  \  the 
stroke,  we  find  the  minimum  limit  of  evaporation  of  water 
required  to  be  about  24  pounds  of  water  per  horse-power 
per  hour. 

Mr.  Charles  E.  Emery  has  experimentally  realized  as  low 
as  39  pounds  of  water  per  horse-power  per  hour  with  the 
given  pressure  in  small  non-condensing  engines. 

If  in  the  last  formula  we  regard  the  product  of  the  pres- 
sure and  the  specific  volume,  PbS,  as  constant,  which  it  is 
with  a  considerable  degree  of  approximation,  we  observe 

T> 

that  the  economy  of  steam  varies  directly  as  the  log  -A 

JB 

showing  that  both  Watt  and  Oliver  Evans  were  partially 
right  in  their  attempts  to  increase  the  economy  of  the  steam- 
engine.  Watt  was  right  in  perfecting  more  and  more  the 
vacuum  obtained,  and  Evans  in  increasing  the  steam-pres- 
sure at  the  boiler.  Neither  of  them  were  wholly  astray, 
but  were  soon  met  and  their  progress  "stopped  by  the  slow 
increase  of  the  economy  beyond  certain  limits,  and  by  the 
practical  difficulties  arising  from  surface-condensation  in 
the  pursuit  of  a  partially  apprehended  law. 

15 


170  THE  RELATIVE  PROPORTIONS 

It  is  evident  that  we  are  limited  as  to  the  steam-pressure, 
and  that  a  great  increase  of  that,  or  a  great  decrease  of 
back  pressure  by  reason  of  more  perfect  vacuums,  will 
render  the  point  of  cut-off  so  early,  the  mean  effective- 
pressure  so  low,  and  the  condensation  so  great  that  the 
necessary  increase  of  volume  in  the  steam-cylinder  will 
prevent  that  compactness  and  concentration  of  power  so 
desirable  in  the  steam-engine ;  and,  although  it  has  already 
been  shown  that  high  rotative  speed  aids  in  rendering  the 
engine  compact,  we  are  also  limited  in  that  direction. 

Regularity  of  speed  is  an  imperative  condition  in  all  en- 
gines used  to  drive  high-speeded  machinery;  under  such 
circumstances  regularity  becomes  the  first  requisite,  and  to 
it  all  other  considerations  must  be  subordinated. 

Mr.  G.  H.  Corliss,  with  his  automatic  cut-off  engine,  was 
the  first  to  satisfactorily  solve  this  problem. 

With  a  constant  steam -pressure  and  a  constant  load  an 
engine  will  run,  under  the  control  of  a  throttling  governor, 
with  an  approximation  to  regularity  determined  by  the  sen- 
sitiveness of  the  governor.  With  a  throttling  governor 
every  change  of  the  steam-pressure,  every  change  in  the 
load,  causes  a  change  in  the  speed  of  the  engine,  no  matter 
how  well  proportioned  it  may  be. 

The  laws  regulating  the  centrifugal  governor  are  such  as 
to  prevent  it  from  ever  becoming  a  very  perfect  mechanism, 
for  the  engine  and  governor  must  go  fast  in  order  to  go  slow, 
and  vice  versd.  With  an  automatic  cut-off  governor  the 
speed  will  be  held  with  greater  regularity  under  all  steam- 
pressures  and  loads  within  the  engine's  capacity,  the  ap- 
proximation to  perfection  attained  being  determined  by  the 
proportions  of  the  fly-wheel  and  the  sensitiveness  of  the 
governor. 

Governors  in  which  the  inertia  of  the  balls  is  utilized  to 
check  any  attempted  change  of  speed,  without  an  apprecia- 


OF  THE  STEAM-ENGINE.  171 

ble  change  of  speed  in  the  governor  itself,  will  ultimately 
supersede  the  centrifugal  governor,  which  is  full  of  radical 
defects. 

Valve-motions  actuated  by  a  weight  or  spring  released 
automatically  by  the  centrifugal  governor  are  comparatively 
so  slow  in  their  motions  as  to  prevent  high  rotative  speeds 
and  short  strokes.  This  form  of  valve-motion  has  been 
elaborated  with  great  skill  and  ingenuity  on  such  engines> 
as  the  Corliss,  Harris-Corliss,  Keynolds-Corliss,  Wheelock, 
and  Brown,  which,  so  long  as  long  stroke  and  slow  rotation 
is  adhered  to,  leave  but  little  to  be  desired  on  the  score  of 
regularity  of  average  speed  or  of  perfection  of  workmanship. 

To  a  different  class  belong  the  Porter- Allen  and  the  Buck- 
eye engines :  in  these  engines,  which  are  the  exponents  of 
high  rotative  speeds  and  short  strokes,  the  cut-off  of  the 
engine  is  actuated  by  means  of  a  link  or  an  eccentric,  whose 
position  is  regulated  by  means  of  a  centrifugal  governor, 
and,  whatever  the  speed,  the  valves  must  keep  pace  with 
the  engine  in  its  motions. 

A  large  number  of  valve-motions  are  in  use  on  the  various 
forms  of  locomotive,  marine,  and  stationary  engines  through- 
out the  civilized  world,  and  it  is  perhaps  worth  our  while  to 
practically  note  the  important  points  of  some  of  the  older 
forms. 

The  plain  hollowed  slide-valve  is  the  most  commonly  used 
on  the  rougher  forms  of  engines,  because  of  its  simplicity ; 
as,  however,  the  earliest  point  at  which  it  can  cut  off  steam 
without  choking  the  exhaust  is  about  f  of  the  stroke,  it  is 
not  economical  with  the  high  pressures  and  large  engines 
of  the  present  day  or  with  the  vacuums  ordinarily  attained 
in  condensing  engines. 

When  a  reversing  gear  is  used  the  Stephenson  link-motion 
is  commonly  used,  where  simplicity  of  mechanism  rather 
than  economy  of  steam  is  a  point  to  be  gained. 


172  THE  RELATIVE  PROPORTIONS 

With  open  rods  it  has  a  variable  lead,  decreasing  from 
the  dead  point  of  the  link ;  with  crossed  rods  it  has  a  varia- 
ble lead,  increasing  from  the  dead  point  of  the  link. 

For  crossed  rods  the  position  of  the  piston  is  nearly  con- 
stant at  the  opening  of  the  steam-ports ;  this  is  a  decided 
advantage.  Crossed  rods  permit  a  slightly  greater  length 
of  eccentric  rods. 

Fastening  the  link  to  the  ends  of  the  eccentric  rods  at  a. 
point  behind  the  centre  line  of  the  link,  as  is  usually  done 
in  American  practice,  introduces  irregularities  into  the 
point  of  cut-off,  but  does  not  affect  the  lead. 

When  the  ends  of  the  eccentric  rods  are  fastened  to  the 
link  at  points  on  the  centre  line  of  the  link,  this  error  does 
hot  occur,  but  larger  eccentrics  are  required,  which  are 
sometimes  inconvenient. 

The  mode  of  suspension  of  the  link  is  far  more  important 
than  is  usually  supposed,  as  imperfect  suspension  not  only 
introduces  serious  errors  in  the  valve-motion,  but  also  affects 
the  durability  of  the  construction  by  producing  excessive 
slip  in  the  block. 

Where  a  constant  lead  is  desired,  using  different  angles 
of  advance  for  the  two  eccentrics  will  produce  a  constant 
lead  in  one  direction  of  motion,  but  ruins  the  action  of  the 
valve-motion  in  the  opposite  direction. 

The  Gooch  link-motion  is  difficult  to  fit  into  an  engine, 
because  of  the  distance  required  between  the  centre  of  the 
crank-shaft  and  the  end  of  the  radius-rod ;  it  is  very  easy 
to  handle,  and  should  therefore  be  used  where  frequent  re- 
versing is  necessary,  as  in  the  cases  of  yard  locomotives  and 
hoisting-engines,  even  though  requiring  considerable  trouble 
to  fit  in. 

If  a  constant  lead  be  a  great  advantage,  this  link-motion 
has  this  advantage.  The  suspension-link  for  the  link  proper 
should  be  made  as  long  as  possible,  and  the  suspension-link 


OF  THE  STEAM-ENGINE.  173 

for  the*  radius-rod  should  be  attached  as  closely  as  possible 
to  the  link-block.  The  tumbling-shaft  should  be  placed  on 
the  opposite  side  of  the  link  from  the  crank-shaft. 

Allan's  link-motion,  having  a  straight  link,  can  be  more 
easily  fitted  up  than  motions  having  a  curved  link ;  it  has 
all  the  disadvantages  of  the  Gooch  link-motion,  but,  unlike 
it,  has  not  a  constant  lead,  and  has  a  greater  number  of 
parts.  Particular  attention  must  be  paid  to  the  relative 
lengths  of  the  tumbling-shaft  arms  for  simultaneously 
raising  and  lowering  the  link  and  radius-rod.  The  Allan 
link-motion  does  not  handle  as  easily  as  the  Gooch. 

Heusinger  von  Waldegg's  link-motion  is  the  easiest  of 
all  to  handle ;  it  can  be  so  arranged  as  to  have  very  little 
slip  in  the  block,  but  cannot  be  easily  attached  to  all  forms 
of  cross-head.  It  is  the  simplest  reversing  gear  yet  men- 
tioned, and  has  a  constant  lead. 

Pius  Fink's  link-motion  is  the  simplest  reversing  gear  in 
existence  which  at  the  same  time  can  be  adjusted  as  a  cut- 
off; it  has  a  constant  lead,  but  a  limited  range  of  action,, 
and  in  this  point  is  inferior  to  all  the  others,  as  well  as 
being  irregular  in  its  variation  from  its  calculated  positions. 
However,  if  carefully  designed,  it  will  do  good  work,  and  is 
very  easy  to  handle  and  very  durable. 

The  fact  that  it  is  impossible  to  avoid  choking  the  ex- 
haust, and  the  consequent  compression  of  the  steam,  thereby 
reducing  the  power  of  the  steam-engine  whenever  the  at- 
tempt is  made  to  cut  off  early  in  the  stroke  with  a  single 
valve,  has  led  engineers  to  devise  double  valve-motions 
which  will  effect  an  early  cut-off  of  the  steam,  independ- 
ently of  the  motion  of  the  hollow  of  the  slide-valve  rel- 
atively to  the  exhaust. 

To  this  type  belong  the  valve-motions  of  the  Porter- Allen* 

*  The  Porter-Allen  valve-motion  is  essentially  Pius  Fink's. 
15* 


174  .     THE  RELATIVE  PROPORTIONS 

and  Buckeye  engines,  both  of  which  show  a  wide  range  of 
action  for  motion  in  one  direction,  and  great  simplicity  of 
structure. 

The  Gonzenbach  link-motion,  with  its  double  steam- 
chest  and  excessively  complicated  outside  gear,  is  now 
almost  obsolete ;  it  will  not  cut  off  sharply  and  well,  ex- 
cept for  very  early  points,  and  that,  too,  only  for  motion  in 
one  direction  ;  and  when  properly  arranged  for  motion  in  one 
direction  gives  a  double  admission  of  steam  for  motion  in 
the  opposite  direction ;  it  is  difficult  to  handle,  and  its  com- 
plication renders  it  very  liable  to  break  down. 

The  Polonceau  link-motion,  while  it  has  not  the  double 
steam-chest  or  as  much  complication  in  its  outside  valve- 
gear,  cannot  cut  off  steam  later  than  ^  of  the  stroke,  and 
is  therefore  not  very  well  adapted  to  general  locomotive  use, 
where  economy  is  sacrificed  to  concentration  of  power.  It 
is  not  very  difficult  to  handle,  but,  like  the  Gooch,  presents 
some  difficulty  in  adapting  to  the  narrow  limits  offered  by 
a  locomotive. 

Meyer's  link-motion  seems  to  have  met  with  the  greatest 
favor  from  builders  of  marine  and  stationary  engines,  as  it 
presents  a  wide  range  for  motion  in  one  direction,  and  a 
greater  range  when  used  as  a  reversing  motion  than  any  of 
the  others.  The  screws  which  move  the  expansion-blocks 
are  liable  to  rust  or  wear,  and  cause  trouble;  the  expan- 
sion-valve should  have  separate  eccentrics  to  work  best ;  it 
is  intricate  in  its  construction,  and  liable  to  break  down. 
It  is  also  difficult  to  handle,  and  absorbs  much  power  unless 
the  distribution-valve  is  balanced. 

An  ingenious  modification  of  the  Meyer  valve-motion  is 
the  Ryder  cut-off,  in  which  a  cylindrical  expansion-valve, 
whose  edges  approach  each  other  after  the  manner  of  an 
isosceles  triangle  wrapped  around  a  cylinder,  is  moved 
around  its  axis  by  means  of  the  governor,  so  as  to  have 


OF  THE  STEAM-ENGINE.  175 

the  same  effect  as  that  of  increasing  or  decreasing  the  dis- 
tance between  the  expansion-blocks  of  the  Meyer  cut-off. 

In  order  that  we  may  make  ourselves  clearly  understood, 
we  recapitulate : 

A  really  good  steam-engine  should  possess  the  following 
qualities :  Concentration  of  power,  economy  of  steam,  regular- 
ity of  speed,  simplicity  of  design,  and  durability  of  construction. 

We  have  endeavored  to  mark  the  limitations  of  the  first 
three  attributes  of  a  good  engine ;  the  last  two  must  be 
worked  out  in  the  shop. 

We  are  perhaps  too  bold  in  our  prediction,  but  it  seems 
to  us  that  the  engine  of  the  future  will  have  a  cylinder 
whose  stroke  is  equal  to  its  diameter;  its  speed  will  be 
limited  by  the  tensile  strength  of  its  revolving  parts ;  its 
bearings  will  be  of  great  length ;  its  cut-off  will  be  deter- 
mined, approximately  at  least,*  by  the  ratio  of  the  back 
pressure  to  the  initial  pressure;  its  valve-motion  will  be 
rigidly  in  connection  with  the  crank-shaft ;  its  reciprocating 
parts  will  be  so  proportioned  as  to  put  a  nearly  constant 
pressure  on  the  crank-pin  ;  its  fly-wheel,  if  it  has  one,  will 
be  proportioned  so  as  not  to  admit  of  more  than  a  prede- 
termined variation  of  speed  during  one  stroke ;  its  governor 
will  no  longer  be  centrifugal,  but  a  revolving  pendulum, 
almost  isochronous,  and  holding  the  engine,  with  all  neces- 
sary approximation,  within  fixed  limitations,  to  a  constant 
speed. 

*  For  reasons  for  varying  cut-off  see  Lecture  XVI.,  on  The  Cheapest 
Point  of  Cut-Off. 


176  THE  RELATIVE  PROPORTIONS 


CHAPTER    XVI. 

(73)  The  Cheapest  Point  of  Cut-Off.— In  the  Journal 
of  the  Franklin  Institute  for  June,  1880,  the  writer  published 
a  brief  paper,  determining,  in  an  approximate  way,  from  a 
purely  dynamical  point  of  view,  the  most  economical  point 
of  cut-off  for  steam  engines.*  This  inquiry  confined  itself 
entirely  to  the  ratio  of  the  indicated  horse-power  to  the  steam 
used,  and  provoked  an  amount  of  criticism  in  German,  Eng- 
lish, and  American  serials  which  was  as  surprising  to  him 
as  the  misapprehension  of  the  limitations  of  the  paper,  to- 
gether with  the  way  in  which  the  intent  of  the  writer  was 
gratuitously  assumed. 

As  a  question  of  finance  the  subject  becomes  more  com- 
plicated, for  the  engine-owner  asks  not,  How  can  I  save  the 
most  steam  ?  but,  All  the  circumstances  being  taken  into 
consideration,  how  can  I  get  the  useful  work  which  I  re- 
quire most  cheaply? 

So  far  as  the  delivery  of  useful  work  by  the  engine  alone 
is  concerned,  a  method  has  been  given  by  Professor  Ran- 
kine,  and  elaborated  by  himself  as  well  as  others;  but, 
perhaps  because  of  insufficiently  profound  study  of  the 
question  from  a  financial  point  of  view,  or  because  the 
subject  was  hardly  deemed  worthy  of  his  thought,  the 
question  was  never  exhausted  by  him  or  his  followers. 

The  ship-owner  says,  How  can  I  obtain  the  power  to 
drive  the  propeller  most  cheaply? 

The  mill-owner  says,  How  can  I  obtain  the  power  to 
drive  the  mill-stones  most  cheaply?  and  the  shop-owner 
says,  How  can  I  obtain  the  power  to  drive  my  machinery 
most  cheaply  ?  and,  as  they  use  the  engine  for  the  purpose 
of  making  money,  wish  to  have  it  designed  for  that  pur- 

*  Appended  to  this  Lecture. 


OF  THE  STEAM-ENGINE.  177 

pose,  and  care  nothing  at  all  for  purely  scientific  considera- 
tions. 

It  is  to  these  men  that  the  writer  will  endeavor  to  make 
reply,  giving  to  a  most  perplexing  question,  involving  many 
considerations,  at  least  an  approximately  correct  reply,  and 
indicating  a  method  which,  by  elaboration  and  a  detailed 
consideration  of  the  thermodynamic  questions  raised,  will, 
we  trust,  enable  an  engineer  to  reach  an  economy  of  useful 
power  as  yet  not  knowingly  obtained  by  other  means  than 
an  inspection  of  the  Profit  and  Loss  account  at  the  end  of 
the  year. 

In  "The  Limitations  of  the  Steam-Engine"  the  writer 
has  stated  as  the  five  points  of  an  engine :  (1)  concentra- 
tion of  power,  (2)  economy  of  steam,  (3)  regularity  of 
speed,  (4)  simplicity  of  design,  and  (5)  durability  of  con- 
struction. 

When  we  do  not  restrict  ourselves  to  economy  of  Nature's 
forces,  economy  of  steam  becomes  economy  of  money. 

The  following  assumptions  are  made  and  particulars  must 
be  understood  in  the  discussion  which  follows. 

The  expansion-curve  of  steam  is  assumed  to  be,  with  suf- 
ficient practical  accuracy,  an  equilateral  hyperbola. 

The  steam  made  by  the  boiler  is  to  the  steam  shown  by 
the  indicator  diagram  as  4  to  3.  This,  certainly,  is  not  cor- 
rect under  all  circumstances,  but  is  an  approximation  de- 
rived from  the  experiments  under  favorable  conditions  upon 
the  Reynolds-Corliss,  the  Harris-Corliss,  and  the  Wheelock 
engines  at  the  Millers'  Exhibition,  Cincinnati,  June,  1880 
(Report  of  J.  T.  Hill,  pages  77  and  79),  and  is  nearly  the 
same  for  both  condensing  and  non-condensing  engines.  Those 
in  possession  of  more  accurate  experimental  data  can  sub- 
stitute other  ratios  in  each  case. 

A  percentage  of  the  true  stroke  must  be  added  at  each 
end  of  the  sketch  which  is  made  to  allow  for  the  clearance, 


178  THE  RELATIVE  PROPORTIONS 

which  must  be  determined.  The  cost  of  all  charges  upon 
the  engine  and  machinery  is  taken  in  steam  for  the  sake  of 
convenience ;  and  this  proceeding  is  perfectly  proper,  since 
money  and  steam  are  convertible. 

For  the  present  all  reference  to  the  saving  of  fuel  result- 
ing from  the  diminished  number  of  heat-units  required  to 
increase  the  pressure  of  steam  is  premeditated ly  omitted, 
because  we  are  practically  limited  by  expense  to  low  press- 
ures in  ordinary  cases. 

The  constant  charges  which  come  upon  engine-boilers  and 
machinery  are  as  follows : 

(1)  Wages  of  attendants  upon  engine,  boilers,  and  ma- 
chinery. 

(2)  Interest  upon  cost  of  engine,  boilers,  and  machinery. 

(3)  Depreciation  of  engine,  boilers,  and  machinery. 

(4)  Repairs  to  engine,  boilers,  and  machinery. 

(5)  Cost  of  lubrication  of  engine,  boilers,  and  machinery. 

(6)  Taxes  and  insurance  upon  engine,  boilers,  and  ma- 
chinery. 

(7)  Interest  upon  cost  of  shelter  and  room  for  engine, 
boilers,  and  machinery. 

Many  other  charges  may  exist  which  the  writer  has  not 
mentioned,  and  some  of  the  charges  mentioned  are  not 
applicable  in  cases  where  methods  of  charging  cost  may 
differ  from  that  of  works  and  factories  engaged  in  the 
production  of  some  staple  article  or  of  a  ship  where  the 
engine  serves  only  for  propulsion.  In  every  case  the  dis- 
tribution of  cost  must  form  an  individual  problem. 

The  only  variable  charge  when  the  engine  is  already 
established  is  the  cost  of  the  steam — i.  e.,  fuel  and  water. 
When  engaged  in  the  design  of  an  engine  and  plant  most 
of  the  constant  charges  may  be  regarded  as  variables,  but 
not  according  to  any  uniform  law,  and  must  be  considered 


OF  THE  STEAM-ENGINE.  179 

as  separate  problems  which  must  be  solved  from  known  pre- 
cedents which  vary  in  different  localities. 

Let  P  =  the  mean  effective  steam-pressure  in  pounds  per 

square  inch. 
"     C  =  the  constant   charges  in  dollars  and  cents  for 

any  assumed  time. 
"    Pb  =  the  absolute  initial  pressure  in  the  cylinder  in 

pounds  per  square  inch. 
"    B  =the  absolute  back  pressure  in  the  cylinder  in 

pounds  per  square  inch  while  the  exhaust-port 

is  open. 
"     e  =  the  fraction  of  the  volume  at  which  steam  is  cut 

off. 

"     V  =the  volume  of  the  steam -cylinder. 
"     c  =  the  factor  of  the  volume  of  steam  proportional  to 

the  ratio  of  the  constant  charges  to  the  cost  of 

steam  for  any  assumed  time. 
"     b  =  the  fraction  of  the  volume  at  which  compression 

begins,  being  measured  from  the  opposite  end 

from  which  e  is  measured. 
"     k  =  the  fraction  of  the  volume  allowed  for  clearance. 


We  can  then  write  : 

Useful  work  PV 


Cost  of  work  in  steam 


B          4  .B 

-e+c_6__ 

b\l 


.  (218) 


But  P=eP6(l  +  nat.  log  -]  - B\  1  -6/1-nat.  log-)  .  (219) 


This  value  of  P  could  be  much  more  accurately  determined 
by  the  careful  use  of  an  indicator. 

Substituting  in  equation  (218),  we  have 


180  THE  RELATIVE  PROPORTIONS 

eP6  (l+nat.  log  ^-) 


Useful  work_ 
Costof  work~4      C 
3      D 


(1  \  /  b  \~\)      B 

1-fnat.  log  —  1 — B    1 — ol  1 — nat.  log  --  I  I  > — o  — 

I  e+  §|eP6  (l+nat.  log  ^  )—  B  [l— 6^1— nat.  log  |j]  l~bp- 

Differentiating  with  respect  to  e  and  seeking  a  maximum, 
we  have 


e  = 


The  natural  logarithm  can  be  obtained  by  multiplying 
the  common  logarithm  by  2.3026,  and  this  transcendental 
equation  must  be  solved  by  a  series  of  approximations,  be- 
ginning with  an  assumption  that 

nat.  log  -  =  nat.  log  — — -,  (222) 


1  -b(l -nat.  log  - 


and  substituting  the  nearer  value  of  -  again  in  the  second 

6 

member  of  equation  (221),  and  so  on  until  two  successive 
values  of  e  nearly  agree. 

As  logarithms  do  not  vary  rapidly,  the  approximations 
required  to  obtain  all  the  accuracy  justified  by  the  data  or 
realizable  in  practice  will  be  few. 

Perhaps  how  to  deduce  the  value  of  c  is  not  clear. 
.  Determine  the  constant  charges,  in  dollars  and  cents, 
upon  engine  and  machinery  for  one  day.  Regardless  of 
the  power,  determine  approximately,  from  the  first  term 
only  of  the  second  member  of  equation  (221),  the  most 
economical  point  of  cut-off  for  steam  alone. 


OF  THE  STEAM-ENGINE.  •  181 

With  the  following  formula  determine  the  weight  of  water 
required  per  horse-power  per  hour  : 

W=  weight  of  water  per  H.-P.  per  hour. 
S  =  specific  volume  of  steam  for  pressure  Pb. 
4 859380e 

<•       r 

1  +  nat.  log- 

6 

(See  "  The  Limitations  of  the  Steam-Engine.") 

With  this  determine  the  cost  of  fuel  and  water  for  the 

required  horse-power  per  day,  remembering  that  £  is  an 

assumed  quantity  which,  at  the  best,  is  only  approximately 

correct. 

The  value  of  c  given   in   formula  (220)  is   deduced  as 

follows : 

De 


c:e::C: 


\  1+nat.  log  - 

L  !l 

Therefore, 

'    (224) 

An  inspection  of  equation  (221)  reveals  many  interesting 
facts. 

We  observe  that  an  increase  of  the  initial  pressure  or  a 
diminution  of  the  back  pressure  renders  the  cut-off  of  steam 
earlier.  Also  that  a  diminution  of  the  back  pressure  has 
much  more  influence  than  an  increase  of  the  initial  pressure. 

If  we  have  no  compression  \b  =  0],  equation  (221)  becomes 

B 


-See  (223)  D  = 


S 

16 


182 


TME  RELATIVE  PROPORTIONS 


If  with  these  conditions  we  substitute  in  equation  (223), 
we  have 

^4        859380 

Pfr£  nat.  log  — 

and  for  any  assumed  initial  pressure  we  can  say, 

1 


W  =  constant: 


nat.  log  — 


The  following  tabulation  will   approximately  show  the 
relative  economy  of  successively  increased  expansions  with 

the  cheapest  point  of  cut- 
off per  horse-power  per 
hour.  At  what  particu- 
lar point  this  gain  is  met 
and  annulled  by  the 
losses  due  to  internal 
condensation  cannot  be 
said  with  our  present  ex- 
perimental knowledge. 
The  demand  for  concen- 
tration of  power,  which 
is  the  greatest  attribute 
of  the  steam-engine,  will 


B 

<=-- 

1 

Saving. 

Per  cent  saved. 

natlog^ 

Succes- 

Of the 

sive 

first 

3.222 

vols. 

volume. 

2.096 

1.226 

37 

37 

1.661 

0.435 

21 

49 

1.431 

0.230 

14 

56 

1.286 

0.145 

10 

60 

1.182 

0.104 

8 

63 

1.107 

0.075 

6 

66 

1.048 

0.059 

5 

69 

T 

7T 

1.000 

0.048 

5 

70 

ft 

0.961 

0.039 

4 

70 

s 

0.927 

0.034 

3 

71 

TV 

0.898 

0.029 

3 

72 

TV 

0.873 

0.025 

2* 

73 

not  permit  us  to  logically  accept  these  results,  as  they  would 
give  very  slight  initial  pressures  for  condensing  engines  or 
require  impossible  points  of  cut-off  with  present  engines. 
A  consideration  of  equation  (223)  will  point  out  to  us  sev- 
eral methods  of  escape.  The  specific  volume  of  steam 
steadily  decreases  as  its  pressure  increases ;  therefore  for  a 
fixed  point  of  cut-off  and  back  pressure  the  economy  of 
steam  effected  is  greater  as  the  pressure  is  greater.  A  re- 
duction of  the  back  pressure  effects  a  rapid  increase  in 


OF  THE  STEAM-ENGINE. 


183 


saving,  since  it  is  multiplied  by  ($)  the  specific  volume, 
usually  a  large  quantity. 

Very  few  condensing  engines  have  good  vacuums,  but  for 
the  purpose  of  giving  approximate  results  we  will  neglect 
the  back  pressure.  Equation  (223)  then  becomes 

constant 


W= 


SP> 


r 


l  +  nat.  log 


IT 


If  the  point  of  cut-off  (e]  is  fixed,  we  can  still  effect  some 
saving  by  increasing  the  steam  pressure,  since  the  product 
SPb  is  a  slowly  increasing  quantity  as  the  initial  pressure  is 
increased. 

If,  again,  we  fix  upon  some  particular  value  of  Pb  in  the 
preceding  equation,  it  becomes 

1 


W  =  constant 


1  -I-  nat.  log  - 


Tabulating  the  results  of  insertion  of  decreasing  values 
of  e,  we  have — 

These  rudely  approxi- 
mate figures  —  which, 
however,  are  probably 
quite  as  accurate  as  any 
that  can  be  realized  in 
practice — show  the  im- 
perative need  that  ex- 
ists for  experimental 
demonstration  of  the 
causes  of  condensation 
in  the  cylinder,  and  the 
application  of  a  remedy  for  it  before  further  intelligent 
progress  can  be  made  in  the  economical  use  of  steam.  A 
comparison  of  the  two  tables  shows  that  in  the  absence  of 


e 

1 

Saving. 

Per  cent  saved. 

1  +  nat.  log  j 

Succes- 

Of the 

sive 

whole 

1 

1 

vols. 

volume. 

i 

0.591 

0.409 

41  • 

41 

1 

0.476 

0.115 

19 

52 

i 

0.419 

0.057 

12 

58 

£ 

0.383 

0.036 

9 

62 

I 

0.358 

0.025 

7 

64 

0.339 

0.019 

5 

66 

| 

0.324 

0.015 

4 

68 

* 

0.312 

0.012 

i| 

69 

A 

0.303 

0.009 

3 

70 

184  THE  RELATIVE  PROPORTIONS 

condensation,  expansion  could  probably  be  carried  further 
with  profit  in  the  former  than  in  the  latter  case.  The  ex- 
pansion necessary  under  existing  conditions  to  effect  the 
greatest  saving  of  steam  per  horse-power  has  already  been 
exceeded  in  many  cases.  Besides  expansion,  the  condensa- 
tion certainly  is  a  function  of  the  temperatures  and  of  the 
surfaces  exposed,  and  of  the  conductivity  of  the  surfaces, 
as  well  as  of  their  time  of  exposure,  and  perhaps  of  the 
relative  time  of  exposure  of  the  interior  surfaces  of  the 
steam-cylinder.  It  should  be  remembered  that  an  increase 
of  horse-power  is  the  only  way  to  decrease  the  constant 
charge  per  horse-power. 

We  have  obtained  but  13  per  cent,  of  the  power  in  coal, 
and  even  now  we  are  at  the  limits  of  commercial  economy 
in  the  use  of  the  steam-engine. 

With  our  present  types  of  boilers  and  engines  it  does  not 
pay  to  use  coal  more  economically.  It  is  cheaper  to  waste 
the  87  per  cent,  of  the  power  of  the  coal  than  to  go  on  in- 
creasing the  cost  of  the  engine  in  the  endeavor  to  save  coal. 

Of  course  if  we  wish  only  to  conserve  Nature's  forces, 
and  disregard  the  money  it  costs  to  produce  a  diminished 
consumption  of  coal,  we  can,  and  probably  will,  succeed  in 
obtaining  larger  results. 

But  money  derives  its  value  from  the  labor  of  humanity, 
and  for  this  reason  should  be  saved  in  preference  to  Nature's 
forces. 

There  are  some  avenues  of  escape  open  to  us  from  this 
dilemma,  and  I  will  mention  them. 

Taking  the  best  type  of  engines  of  to-day  as  a  starting- 
point,  we  must  depart  in  the  following  directions : 

We  do  not  particularly  need  to  increase  the  efficiency  of 
the  boiler  as  an  evaporator,  but  we  must  increase  its  ability 
to  withstand  pressure  without  increasing  its  cost. 

We  must  decrease  the  condensation  inside  of  the  steam- 


OF  THE  STEAM-ENGINE.  185 

cylinder  by  using  a  non-conducting  surface  or  by  superheat- 
ing to  a  small  extent. 

We  must  decrease  the  friction  of  the  engine  and  of  the 
machinery  of  transmission  to  the  point  where  the  useful 
work  is  delivered. 

We  must  produce  better  vacuums  in  the  condenser,  and 
diminish  its  cost. 

We  must  diminish  the  cost  of  the  engine. 

We  must  diminish  the  cost  of  the  attendance  on  engines, 
boilers,  and  machinery,  and  of  lubrication. 

We  must  increase  the  durability  of  engines,  boilers,  and 
machinery. 

Coal  is  too  cheap  even  now  to  admit  of  increased  economy 
of  it  at  the  cost  of  increased  outlay  for  plant  and  attendance. 
We  would  be  saving  coal  without  saving  money,  or  rather 
spending  more  money  in  the  difference  in  constant  charges, 
such  as  interest,  deterioration,  and  attendance  are,  than  we 
would  save  in  the  difference  in  the  coal-bill. 

The  reason  why  automatic  cut-off  engines  have  produced 
such  favorable  results  is  this :  The  condition  in  every  case 
is  that  a  point  of  cut-off  shall  be  determined,  and  then  the 
engine  designed  for  some  certain  power  with  that  cut-off. 
The  demands  of  the  shop  or  mill  will  not  permit  such  a 
condition  of  uniformity  to  be  fulfilled ;  and  here  the  automatic 
cut-off  steps  in,  and,  with  the  added  advantage  of  its  carefully 
regulated  speed,  uses  less  steam,  although  not  so  economically, 
per  horse-power  per  hour,  whenever  the  demand  for  power 
is  lessened ;  and  so  at  the  end  of  the  month  a  real  saving  is 
apparent  in  the  coal-bill,  although  the  engine  may  not  have 
cut  off  at  the  point  of  least  cost  for  a  good  part  of  the  time. 

The  engine-owner  must  have  an  engine  which  is  adequate 
to  the  greatest  demand  for  power  which  may  be  made  upon 
it,  but  he  should  not  choose  an  engine  which  would  cut-off 
too  early  under  its  average  load. 

16* 


186  THE  RELATIVE  PROPORTIONS 

Economy  of  Steam  Alone  without  Compression.     [From 

J.  F.  I.,  June,  1880.] 

Let  P  =  the  mean  pressure  in  pounds  per  square  inch. 
"    P6  =  the  initial  pressure  in  pounds  per  square  inch 

(absolute). 
"    B  =  the  back  pressure  in  pounds  per  square  inch 

(absolute). 
"     e  =the  fractional  part  of  the  stroke  at  which  the 

steam  is  cut  off  by  the  valve-motion. 
"     V  =  the  volume  of  the  steam-cylinder. 
"    E  =  the  economy. 

Now,  it  will  further  be  acknowledged  that  the  economy 
increases  directly  as  the  work  done  during  one  stroke  and 
inversely  as  the  steam  used  during  one  stroke.  Therefore 
we  have 

PV    P 

*-%'*•  (225) 

We  have  further  the  well-known  formula  for  the  mean 
effective  pressure  of  steam  used  expansively,  supposed  to 
expand  according  to  Mariotte's  law: 

r     fi 

P=eP6  \1+L- \-B.  (226) 

L    e] 

While  this  does  not  accurately  represent  the  law  of  ex- 
pansion of  steam,  it  does  it  with  close  approximation  for  all 
practical  purposes.  Substituting  the  value  of  P  from  equa- 
tion (226)  in  equation  (225),  we  have 


E=Pb  1-t-i-  --.  (227) 

L       ll     6 

Differentiating  with  respect  to  e,  and  seeking  the  max- 
imum, we  find  it  to  be 

e-2-.  (228) 


OF  THE  STEAM-ENGINE.  187 


CHAPTER    XVII. 

(74)  The  Errors  of  the  Zeuner  Diagram  as  Applied 
to  the  Stephenson  Link-Motion.*  —  I.  INTRODUCTION. 
The  mathematical  elegance  of  Professor  Gustav  Zeuner's 
Treatise  on  Valve-  Gears  is  due  to  the  fact  that  he  has  shown 
that  the  equation  representing  the  distance  of  a  slide-valve, 
controlled  by  an  eccentric  or  by  means  of  a  link,  is  in  all 
cases  with  greater  or  less  approximation  the  polar  equation 
of  a  circle.  Deservedly,  his  work  has  met  with  a  most 
gratifying  acceptance  from  all  intelligent  engineers,  as  not 
only  being  the  most  correct,  but  also  the  only  method 
which,  without  the  aid  of  models  or  templates,  enables 
the  practitioner  to  devise  and  study  any  desired  form  of 
valve-gear. 

As  shown  in  the  drawing  (Fig.  32),  upon  the  top  of  a 
standard  behind  the  section  of  the  cylinder  a  pulley  of 
equal  size  with  the  crank-shaft  was  connected  with  the 
crank-shaft  by  means  of  a  steel  saw-band  running  upon 
thin  gutta-percha  strips  glued  to  the  shaft  and  pulley  sur- 
faces. This  steel  band  was  kept  very  taut  by  means  of  a 
stretching  pulley  about  the  middle  of  its  length. 

Upon  the  end  of  the  pulley-shaft  and  just  back  of  the 
valve  a  drawing-board  was  so  attached  as  to  permit  a  pen- 
cil, attached  to  the  valve  and  kept  pressed  against  the  paper 

*  The  drawings  for  this  chapter  were  made  by  Mr.  G.  H.  Lewis,  a 
graduate  of  the  department  of  Dynamical  Engineering,  and  used  by 
him  as  a  part  of  a  thesis. 

The  mathematical  treatment  is  my  own.  Mr.  Lewis's  drawings 
have  been  somewhat  added  to,  in  order  to  give  graphical  methods  of 
determining  the  errors  of  the  diagram.  I  am  indebted  to  Mr.  Lewis 
for  many  ingenious  and  thoughtful  suggestions  and  much  accurate 
and  painstaking  work  in  tracing  the  diagrams. 


188 


THE  RELATIVE  PROPORTIONS 


stretched  upon  the 
drawing  -  board,  to 
trace  the  curve, 
showing  the  distance 
of  the  valve  from  its 
central  position. 

Had  the  drawing- 
board  been  attached 
directly  to  the  crank- 
shaft, and  a  rod  hav- 
ing a  pencil  in  the 
end  been  attached 
to  the  link-block  or 
any  point  on  the 
valve  or  valve-stem, 
and  carried  back  to 
the  centre  of  the 
board,  it  would  have 
been  more  service- 
able for  scientific 


purposes,    as 


elimi- 


nating some  of  the 
possible  sources  of 
error  due  to  the  im- 
perfections of  the 
model. 

This  model  was 
constructed  of  iron, 
brass  and  mahog- 
any, and  every  pos- 
sible precaution  was 
taken  to  obtain  rigid- 
ity and  avoid  shrink- 
age ;  it  was  construct- 


OF  THE  STEAM-ENGINE. 


189 


ed  full  size  from  the  dimensions  stated  by  Zeimer  in  his 
Treatise  on  Valve -Gears,  page  78. 

Eccentricity  =  r  =  2.36  inches. 

Angular  advance  =  3  =  30°. 

Length  of  the  eccentric-rods  =  I  =  55.1  inches. 

Half  length  of  the  link  =  c  =  5.9  inches. 

Outside  lap  =  e  =  0.94  inches. 

Inside  lap  =  i  =  0.27 

Open  eccentric-rods  and  equal  angles  of  advance  were 
taken.  The  link  was  so  attached  to  the  eccentric-rods  as 
to  permit  the  link-block  to  be  placed  immediately  in  front 
of  the  ends  of  the  eccentric-rods ;  in  other  words,  so  that 
the  variable  distance  u  of  the  link-block  from  the  centre  of 
the  link  could  at  its  maximum  be  made  equal  to  the  half 
length  of  the  link  c. 

This  form  of  link  is  shown  in  Fig.  33  a. 


FIG.  33. 


(a) 


\ 


190  THE  RELATIVE  PROPORTIONS 

The  diagrams  taken  upon  this  model  clearly  showed  that 
some  greater  sources  of  error  existed  than  the  so-called 
"Missing  Quantity"  of  Zeuner. 

Acceptance  of  authority  is  a  great  preventive  of  advance- 
ment of  knowledge,  and  it  will  be  our  task  to  show  clearly 
what  points  have  been  overlooked  by  Professor  Zeuner,  with, 
we  hope,  the  result  of  making  even  more  clearly  understood 
this  construction,  so  simple  in  its  mechanism  and  so  intricate 
in  its  action. 

II.  THE  SIMPLE  SLIDE-VALVE.  CONSIDERATION  OF 
THE  MISSING  QUANTITY  IN  THE  SIMPLE  SLIDE-VALVE. 
SETTING  VALVE  FOR  EQUAL  LEADS  EQUIVALENT  TO  AL- 
TERING THE  LAPS  OF  THE  VALVE.  For  the  sake  of  sim- 
plicity let  us  first  consider  the  simple  slide-valve. 

On  page  11  of  his  Treatise  on  Valve- Gears,  Zeuner  gives 
for  the  distance  of  a  simple  slide-valve  from  its  centre  of 
motion  £, 

'   £  =  rsm(w  +  d')+  —  sin (2#  +  w) sin w.       (229) 

The  first  term  of  the  second  member  of  this  equation  is  the 
polar  equation  of  a  circle,  with  the  origin  in  its  circumfer- 
ence and  its  diameter  forming  an  angle  equal  to  <5;  with 
the  axis  of  ordinates  OY  (see  Fig.  34)  w  is  the  angle 
which  the  crank  forms  with  the  axis  of  abscissas  OX. 
All  of  this  can  readily  be  understood  from  the  explana- 
tions given  in  the  book. 

It  is  with  the  second  term  of  the  second  member  ("  the 
missing  quantity  ")  that  we  shall  have  particularly  to  deal, 
for  Zeuner  has  considered  it  as  inappreciable  in  most  cases ; 
which  is  not  practically  true,  for  many  cases  occur  in  which 
of  necessity  the  eccentric-rods  are  comparatively  short. 

Dr.  Zeuner  fixes  the  central  position  of  the  slide-valve  by 
taking  the  mean  of  the  two  positions  of  the  valve  when  the 


OF  THE  STEAM-ENGINE.  191 

crank  is  on  its  dead  points.  He  does  this  on  the  assumption 
that  the  valve  will  be  set  for  equal  leads ;  which  is  always  the 
proper  method. 

This  central  position  differs  from  the  true  central  position 

r2  cos2  <5 
by  a  quantity  = (230),  for  the  true  central  point  of 

the  valve  travel  is  a  mean  between  the  extreme  positions  of 
the  valve,  and  farther  away  from  the  crank-shaft,  a  distance 
equal  to  the  above-stated  quantity ;  therefore  at  one  extreme 

7*2  COS2  S 

the  valve's  distance  from  Zeuner's  centre  =  r  + ,  and 

r2  cos2  d 
at  the  other  extreme  =  r . 

If  now  we  can  convert  the  missing  quantity  into  a  func- 
tion of  the  theoretical  valve  distance  from  its  centre  for 
equal  leads  (Zeuner's  centre),  we  can  much  more  conve- 
niently lay  down  the  irregular  curve  of  the  valve-circle  for 
the  case  of  a  short  eccentric-rod. 

According  to  the  diagram,  £  =  rsin  (w  +  <5)  (231).  Page 
43,  Z.  T.  V.  G.*  the  missing  quantity  is  given  as 

z  -  —  [cos2  d  -  cos2  (w  +  <?)],  (232) 

Zv 

T\         S*        -O         ^  Sin2  (W  +  ^)  ^OQON 

or  2  =  _(cos2<5-l)  + |p — S         (233) 

or  2lz  =  r2  (cos2  d-l)  +  r2  sin2  (w  +  d).         (234) 

Letting  0=r2(cos*<5-l)  and  substituting  £  for  its  value, 
we  have 

e  -2&--C,  (235) 

the  equation  of  a  parabola  whose  ordinates  are  the  theoret- 
ical travels  of  the  valve  from  its  centre  of  motion,  and 
whose  abscissas  are  the  missing  quantities  for  the  same. 

*  Abbreviation  of  Zeuner's  Treatise  on  Valve-Gears. 


192  THE  RELATIVE  PROPORTIONS 

The  radius  of  curvature  of  this  parabola  at  its  vertex 
=  %  the  latus  rectum  or  parameter,  and  is  equal  to  /,  the 
length  of  the  eccentric-rod,  and  we  can  substitute  an  arc 
of  a  circle  with  the  radius  I  for  this  parabola  without  ap- 
preciable error. 

For  the  travel  £  =  o, 

z  =  —  -T  —  sin2  5.  (236) 

cyi          '    nj 

Forf  =  r, 

z=  i  —  cos2  d.  (237) 

For  2  =  0, 

*    2  >          *     2  /  V\  ^OQQ"\ 

Therefore  w  =  o. 

That  is  to  say,  the  "  missing  quantity  "  disappears  on  the 
dead  points,  since  the  valve  is  actually  set  for  different  leads. 

To  lay  down  the  actual  curves  of  valve  travel,  the  "  miss- 
ing quantity  "  being  taken  into  account. 

Fig.  34. — With  a  radius  0  L0  and  the  centre  O  describe 
an  arc  L0L  to  intersection  L  with  the  diameter  of  the  valve 
circle  0  P0.  At  the  point  O,  and  at  right  angles  with  0  P0, 
draw  the  indefinite  line  O  Z. 

With  a  radius  of  compass  =  I,  and  with  the  centre  on  the 
line  O  Z,  describe  through  L  and  Lj  the  arc  Q  L  L^.  The 
ordinates  to  this  arc  from  the  line  O  P0  measure  in  quantity 
and  direction  the  values  of  the  "  missing  quantity,"  which 
must  be  added  to  or  subtracted  from  the  theoretical  radius 
vector  in  order  to  obtain  the  true  curve  of  the  motion  of 
the  valve. 

Fig.  34,  for  the  purpose  of  showing  an  extreme  case,  has 
been  laid  down  to  scale  as  follows : 

Eccentricity  =  r  =  2  inches. 
Angular  advance  =  3  =  30°. 


OF  THE  STEAM-ENGINE. 


FIG.  34. 


193 


H,  H,          R 


H*     H        H,    R| 


Outside  lap  =  e  =  0.82  inches. 
Width  of  port  =  a  =  0.75  inches. 
Length  of  eccentric-rod  =  I  =  8  inches. 
The  construction  for  the  missing  quantity,  for  the  sake  of 
clearness,  has  all  been  added  in  heavy  lines. 
The  effect  of  the  "  missing  quantity/'  when  considerable 
17 


194  THE  RELATIVE  PROPORTIONS 

enough  to  be  noticed,  is,  when  the  piston-head  moves  toward 
the  crank-shaft,  the  cylinder  being  at  the  right  hand, 

(1)  To  delay  slightly  the  pre-admission  of  steam. 

(2)  To  increase  the  over-travel. 

(3)  To  hasten  the  cut-off  of  the  steam  (very  slightly). 

(4)  To  hasten  the  compression  of  the  steam. 

(5)  To  hasten  the  release  of  the  steam. 

When  the  piston-head  moves  away  from  the  crank-shaft, 

(1)  To  hasten  the  pre-admission. 

(2)  To  diminish  the  over-travel. 

(3)  To  delay  the  cut-off  (very  slightly). 

(4)  To  delay  the  compression. 

(5)  To  delay  the  release. 

A  glance  at  the  diagram  at  once  reveals  the  fact  that 
equalizing  the  lead  very  nearly  equalizes  the  cut-off. 

It  is  only  when  the  valve  is  set  for  equal  extreme  travels 
from  the  centre  that  different  laps  are  required.  No  atten- 
tion has  been  paid  to  the  variation  in  position  of  the  piston 
due  to  the  obliquity  of  the  connecting-rod. 

III.  THE  PISTON'S  POSITION.  The  effect  of  the  obliquity 
of  the  connecting-rod  is  to  keep  the  piston  nearer  to  the 
crank-shaft  when  it  is  moving  away  from  it,  and  to  draw  it 
closer  to  the  crank-shaft  when  it  is  moving  toward  it,  than 
it  would  be  if  the  connecting-rod  was  constantly  parallel  to 
the  centre  line  of  the  cylinder. 

At  the  dead  points,  the  connecting-rod  being  in  the  centre 
line  of  the  cylinder,  this  action  ceases. 
Letting  w  =  angle  of  the  crank, 
"       R  =  radius    " 

"       L  =  the  length  of  the  connecting-rod, 
we  would  have,  if  the  connecting-rod  were  constantly  par- 
allel to  the  centre  line  of  the  cylinder,  for  the  space  passed 
over  by  the  piston-head  =  S, 

;  (239) 


OF  THE  STEAM-ENGINE.  195 

and  when  we  take  the  obliquity  of  the  connecting-rod  into 
consideration, 


T  N  A.      VL1  —  R*  sin2  w\   t 
-L}ll  -----  1.  (240) 


Then  for  the  difference  d  between  the  two  positions  we 
have 


(241) 

or,  expanding, 

7O2 

d  =  --  sin2  w,  approximately.  (242) 

2Z/ 

Fig.  34.  —  The  positions  HI  to  H7  can  be  corrected  by  lay- 
ing down  in  the  opposite  direction  from  the  cylinder,  from 
the  points  as  already  found,  the  values  of  d. 

It  will  be  observed  that  the  equation  for  d  is  the  equation 
of  a  parabola  whose  se,mi-latus  rectum  is  equal  to  L.  Fur- 
ther, for  w  =  o  or  180°  d=o.  If  for  this  parabola  we  sub- 
stitute the  osculatory  circle  of  a  radius  L  to  its  vertex,  we 
are  practically  close  enough. 

If  now  with  a  radius  of  compass  =  L,  with  one  point  in 
M  and  the  other  on  the  line  Y  O  bisecting  the  cylinder,  we 
describe  the  arc  K2K3,  we  have,  with  sufficient  approxima- 
tion, the  desired  parabola. 

Taking  off  for  the  position  O  R  of  the  crank  the  distance 
JRT=Rsmw,  and  laying  it  off  from  M  to  S,  we  have  the 
correction  S  U  of  the  position  of  the  piston  H,  which,  if  we 
consider  the  cylinder  at  the  right-hand  side,  should  be  laid 
off  to  the  left  of  H,  giving  the  true  position  of  the  piston- 
head  at  H'. 

Thus  we  can  lay  down  graphically  the  actual  positions 
of  the  piston  and  the  true  distances  of  the  slide-valve  from 
its  centre  of  motion,  when  set  for  equal  leads  for  every  posi- 
tion of  the  crank  and  for  any  proportions  of  the  mechanism. 


196  THE  RELATIVE  PROPORTIONS 

For  the  sake  of  emphasis  we  again  repeat :  Different  laps 
are  not  necessary  when  the  valve  is  set  for  equal  leads,  when 
the  piston  position  is  disregarded. 

Altering  the  laps  will  alter  the  leads.  If  the  piston 
position  is  regarded  and  the  alteration  in  the  leads  is  dis- 
vegarded  for  the  sake  of  a  very  accurate  cut-off,  the  lap 
should  be  shortened  on  the  side  toward  the  crank  and 
lengthened  on  the  side  away  from  the  crank.  These 
amounts  can  be  determined  from  the  diagram. 

It  is  only  in  the  case  of  a  very  short  connecting-rod  that 
such  a  procedure  is  necessary ;  short  eccentric-rods  do  not 
require  it. 

IV.  THE  STEPHENSON  LINK-MOTION.  ERROR  DUE  TO 
AN  IMPERFECT  MODE  OF  ATTACHING  THE  LINK  TO  THE 
ECCENTRIC-KODS.  On  pages  56-98  of  Z.  T.  V.  G.  the  Ste- 
phenson  Link-Motion  is  very  fully  treated  for  both  open  and 
crossed  rods,  and  for  both  forms  of  link  shown  in  Fig.  33  a 
and  b,  no  distinction  being  made  between  them. 

In  Fig.  33  a  it  will  be  observed  that  the  rods  are  attached 
on  the  concave  side  at  the  points  C  and  d,  introducing  an 
error  which  we  will  next  endeavor  to  determine. 

Fig.  35  is  the  Zeuner  diagram  carefully  laid  down  for  the 
dimensions  already  given  of  the  model,  on  which  was  used  a 
form  of  link  shown  in  Fig.  33  a. 

The  method  of  making  the  slide-valve  describe  its  own 
diagram  has  already  been  explained.  It  is  only  necessary 
to  add  that  as  the  drawing-board  turns  synchronously  with 
the  crank,  the  valve-circles  (curves)  will  both  be  on  the 
same  side  of  the  origin  instead  of  on  opposite  sides,  as 
drawn  for  the  sake  of  clearness  in  Fig.  34. 

The  object  of  making  the  link  of  the  form  Fig.  33  a  is 
twofold :  first,  to  reduce  the  eccentricity ;  second,  to  enable 
us  to  place  the  valve  wholly  under  the  control  of  one  eccen- 
tric-rod. 


OF  THE  STEAM-ENGINE. 
FIG.  35. 


197 


-*» 


Fig.  36  is  a  centre-line  sketch  of  Fig.  33  a,  similarly  let- 
tered.   It  will  be  observed  that  as  the  suspended  link  sweeps 


17  * 


198  THE  RELATIVE  PROPORTIONS 

to  and  fro  with  a  scythe-like  motion,  the  line  K  C  forms  an 
angle  with  the  horizontal  line  K  N1?  which  is  approximately 
equal  to  the  angle  a,  which  the  chord  of  the  link  forms  with 
the  vertical. 

The  value  of  sin  a  is  given  with  very  close  approximation 
on  page  61,  equation  (11),  Z.  T.  V.  G. 

As  our  only  object  is  to  point  out  an  error  which  can  be 


avoided,  we  will  make  use  of  the  principal  term  of  this 
quantity,  and  take 

sin  a  =  -  cos  d  sin  w.  (243) 

c 

Let  us  denote  the  missing  quantity  due  to  this  error  by  zl  = 
NNlt  its  effect  being  to  keep  the  link  closer  to  the  crank- 
shaft except  where  it  equals  zero. 

Let  KC=q:  

NN,  =  zl  =  q  (1  -  cos  «)  -  2/1  -  ^  1  -  *-  cos2  d  sin2  w \  (244) 

or,  expanding  the  quantity  under  the  radical,  and  neglecting 
terms  containing  greater  than  the  second  power  of  the  cir- 
cular functions,  we  have 

z^^-cos^sin2™.  (245) 

2c3 


OF  THE  STEAM-ENGINE.  199 

For  w  =  90°  this  quantity  is  a  maximum,  and  for  w  =  0°  it 
is  equal  to  zero.  That  is,  it  does  not  appear  in  the  lead 
when  the  valve  is  set  for  equal  leads,  but  it  does  attain  its 
maximum  near  the  point  of  usual  cut-off,  and  is  particularly 
pernicious  there  and  at  the  point  of  exhaust  closure.  The 
reason  that  it  has  remained  unperceived  hitherto  is  probably 
because  it  does  not  appear  in  the  lead. 
Transposing,  we  have 

0/.2 

(246) 


q  cos2  3 

The  equation  of  a  parabola  whose  ordinates  are  rsmw 

c2 

and  whose  abscissas  are  zit  its  semi-latus  rectum  is  -  —  » 

q  cos  o 

which  is  also  the  radius  of  curvature  of  the  osculatory  circle 
to  its  vertex. 

A  moment's  reflection  will  convince  the  reader  that  the 
error  due  to  Zeuner's  missing  quantity  is  inappreciable 
(where  of  any  consequence)  in  the  present  case.  See  Fig. 
34  and  explanation. 

To  determine  the  error  z1?  through  O  (Fig.  35),  with  a 
centre  on  O  X  produced,  describe  an  arc  of  a  circle  T  0  V 

with  a  radius  =  -  .    With  0  as  a  centre  and  the  radius 
q  cos2  3 

r  describe  an  arc  S  T  U.  Draw  any  position  of  crank  as 
O  Z  to  intersection  Z  with  the  arc  S  T  U.  Parallel  to  O  X 
draw  through  Z  the  line  Z  Q.  The  distance  PQ  -  the  error, 
which  can  be  laid  off  both  inside  and  outside  the  theoretical 
valve-circle,  as  at  pfpb.  In  the  model  q  =  3  inches,  3  =  30°, 
and  c  =  5.9  inches. 
Therefore 

c2 

—  —  15.47  inches, 
q  cos2  3 

which  is  the  radius  of  the  arc  V  0  Q  T. 


200 


THE  RELATIVE  PROPORTIONS 


Laying  down  after  the  manner  described  the  arcs  blc  fh, 
we  have  the  corrected  circles  for  the  valve-motion  at  the 
IV  grades.  These  arcs  are  laid  down  for  the  neighborhood 
of  the  point  of  cut-off  only. 

This  most  pernicious  error  can  be  avoided  by  the  use  of 

FIG.  37. 


the  link  (Fig.  33  5),  although  a  larger  eccentric  is  required, 
and,  therefore,  it  is  sometimes  difficult  to  fit  into  confined 


OF  THE  STEAM-ENGINE.  201 

spaces.     Certainly  it  is  of  great  importance  to   avoid   so 
faulty  a  construction  if  it  be  possible. 


FIG.  38. 


Figs.  37  and  38  are  diagrams  automatically  traced  by  the 
working  model. 

To  avoid  the  errors  due  to  suspension,  the  link-block  was 
clamped  in  the  link  for  each  grade,  and  the  link,  therefore, 
swung  upon  the  rocker-shaft  arm. 

To  a,void  the  errors  due  to  the  "  lost  motion,"  the  valve- 
circles  were  traced  twice  by  reversing  the  direction  of  the 
motion,  and  the  mean  between  the  two  circles  traced  with 


202  THE  RELATIVE  PROPORTIONS 

pen  and  ink  by  hand.  The  difference  was  very  slight,  if 
any  at  all. 

It  will  be  seen  that  these  actual  valve-curves  verify  with 
great  accuracy  the  corrected  valve-circles  (Fig.  35)  for  the 
fourth  grade. 

Similar  corrections  can  be  made  for  each  grade  of  link  if 
desired. 

The  letters  /  and  b  refer  to  the  direction  of  motion  of 
the  piston-head,  forward  (/)  meaning  toward  the  crank- 
shaft, and  backward  (6)  meaning  away  from  the  crank- 
shaft, a  rocker-shaft  intervened  reversing  the  direction  of 
motion  of  the  valve. 

When  the  form  of  link  shown  (Fig.  33  6)  is  used,  the 
increased  eccentricity  required  will  increase  the  "missing 
quantity  "  given  by  Prof.  Zeuner,  and  it  must  therefore  be 
guarded  against,  particularly  in  extreme  cases. 

Cases  may  occur  when  it  will  prove  advantageous  to  at- 
tach the  eccentric-rods  to  the  link  at  points  nearer  its  centre 
than  the  extreme  limits  of  the  travel  of  the  link-block,  but 
special  pains  should  be  taken  to  place  the  centre  of  the  pin- 
joint  on  the  central  arc  of  the  link ;  this  method  of  attach- 
ment, however,  will  result  in  increasing  the  slip  of  the  link- 
block. 

V.  SLIP  OF  THE  LINK-BLOCK.  Zeuner  gives  two  cases 
of  the  suspension  of  the  link,  by  means  of  a  hanger  attached 
at  the  centre  of  the  chord  of  the  link  and  at  the  bottom  of 
the  link ;  in  the  first  case  the  upper  end  of  the  hanger 
should  theoretically  move  in  an  arc  of  a  circle  which  has 
for  a  radius  the  length  of  the  eccentric-rod,  and  whose 
centre  is  above  the  centre  line  a  distance  equal  to  the 
length  of  the  hanger.  The  lower  end  of  the  hanger 
should  be  attached  at  the  centre  of  the  link  and  on  the 
central  arc  of  the  link,  thus  placing  the  origin  of  the  arc 
of  suspension  at  a  horizontal  distance  equal  to  the  length 


OF  THE  STEAM-ENGINE. 


203 


of  the  eccentric-rod  from  the  centre  of  the  crank-shaft. 
Fig.  39  will  render  this  clear.  Jlt  and  not  J,  should  be 
the  point  of  attachment  of  the  hanger,  and  L,  not  K1? 


FIG.  39. 


should  be  the  centre  of  the  arc  in  which  the  upper  end 
of  the  hanger  E  should  move. 

"We  can  thus  avoid  increasing  the  slip  by  the  quantity 
J  Jx  tan  a  in  one  direction,  and  decreasing  it  by  the  same 
amount  in  the  other  direction. 

Fig.  40  shows  the  second  method  of  suspension  of  the 
link  by  a  hanger  attached  to  the  bottom. 

Both  of  these  methods  are  fully  explained  by  Zeuner, 
and  the  reader  is  referred  to  his  work  for  further  details. 

It  will  be  perceived  that  the  hanger  is  but  a  rude  ap- 
proximation to  a  parallel  motion,  used  only  because  of  its 
simplicity  and  lightness  when  the  link-block  is  forced  to 
move  to  and  fro  in  a  straight  line. 

When  the  link-block  is  attached  to  the  end  of  a  rocker- 
shaft  arm,  as  is  commonly  the  case  for  American  locomo- 


204 


THE  RELATIVE  PROPORTIONS 


tives,  if  the  hanger  is  made  the  same  length  as  the  rocker- 
shaft  arm,  there  will  be  no  slip  when  the  link-block  coincides 
with  the  point  of  suspension ;  the  slip  for  other  positions  of 
the  link-block  will  be  due  to  the  angular  position  of  the 
link. 

Assuming,  when  the  block  is  forced  to  move  in  a  straight 
line,  that  some  method  has  been  adopted  to  force  the  point 
of  suspension  of  the  link  to  move  in  an  at  least  very  close 

FIG.  40. 


approximation  to  a  straight  line,  and,  further,  that  when 
the  link-block  moves  in  an  arc  of  a  circle  of  a  given  radius 
the  hanger  is  of  the  same  length  as  this  radius,  we  can 
consider  the  slip  as  due  only  to  the  angular  position  of 
the  link. 

Of  course  these  conditions  cannot  always  be  fulfilled,  but 
it  is  best  to  know  what  ought  to  be  done,  even  if  we  cannot 
exactly  do  it. 

Fig.  41  shows  the  two  positions  of  the  links  K  Kx  and 
K,K3,  for  which  the  slip  is  zero ;  and  letting  8  equal  the 


OF  THE  STEAM-ENGINE.  205 

amount  of  slip  for  all  other  positions,  if  we  suppose  the 
hanger  attached  to  the  middle  point  of  the  link,  and  u  the 
distance  of  the  link-block  from  that  point,  we  will  have 

s  =  (seca-l)w,  (247) 

or,  since  the  angle  a  is  always  very  small, 

S  =  u(l-COSa)=u(l-Vl-BUL*a);        (248) 
and  substituting  for  sin  a  its  value,  and  expanding  and  neg- 

FIQ.4L 


lecting  all  terms  containing  higher  powers  than  the  square 
of  the  circular  functions,  we  have 


= cos  o  sin*  10. 


(249) 


We  thus  see  that  the  effect  of  the  slip  does  not  appear  in 
the  lead,  but,  being  a  maximum  for  w  =  90°  or  270°,  will 
affect  the  points  of  cut-off  and  exhaust  closure. 

Increasing  the  angular  advance  diminishes  the  slip,  as 
also  does  increasing  the  length  of  the  link.  The  tendency 
of  the  slip  is  to  increase  the  travel  of  the  valve  by  an 
amount  V : 

.  (260) 


18 


206 


THE  RELATIVE  PROPORTIONS 


This  amount  is  very  small  for  a  well-proportioned  valve- 
gear,  but  it  increases  directly  as  the  distance  u  from  the 
point  of  attachment  of  the  hanger  to  the  link. 


FIG.  42. 


u=o 


When  the  link  is  suspended  from  the  bottom  the  value  w 
must  be  replaced  by  (c  +  u). 

We  thus  see  that  for  general  usage  at  all  points  suspend- 
ing the  link  at  the  middle  is  the  best,  while  if  one  particular 
point  is  intended  to  be  constantly  used,  and  the  other  points 
only  exceptionally,  it  is  best  to  attach  the  hanger  to  the  link 
at  that  point. 

If  the  tumbling-shaft  arm  cannot  be  made  equal  to  the 
length  of  the  eccentric-rods  (and  for  obvious  reasons  it 
rarely  can  be  so  proportioned),  the  centre  of  the  tumbling- 
shaft  must  be  so  placed  as  to  make  an  arc,  struck  with  its 
arm  as  a  radius,  intersect  the  theoretical  arc  at  the  point  or 
points  of  greatest  usage. 

From  what  has  been  said  about  the  position  of  the  arc  of 
suspension,  it  will  readily  be  perceived  that  its  length  on 
either  side  of  the  horizontal  line  E0K!  (Figs.  39  and  40)  is 
determined  by  the  point  of  attachment  of  the  hanger  to  the 
link. 


OF  THE  STEAM-ENGINE. 


207 


The  link-motion  on  which  experiments  were  made  with  a 
view  to  testing  the  correctness  of  these  results  had  the  fol- 
lowing dimensions : 

Length  of  eccentric-rods  =  I  =  18  inches. 
Kadius  of  eccentricity  =  r  =  1£         " 
Length  of  link  =  2c  =  6  " 

Angular  advance  —  d  =  30°. 
Open  rods. 
These  results  verify  the  above  theory  only  in  a  qualitative 

FIG.  43. 


«*-> *e 


way,  as  the  upper  end  of  the  hanger  was  not  always  kept  on 
the  true  arc  of  suspension. 

The  link  being  suspended  after  the  manner  described,  a 
pencil  was  fixed  in  the  link-block,  and  the  block  success- 
ively fixed  at  different  grades,  the  pencil  being  allowed  to 
trace  on  a  paper  back  of  it  the  curves  of  slip. 

CASE  I.   The  link  suspended  at  the  centre  of  its  arc. — Fig. 


208 


THE  RELATIVE  PROPORTIONS 


42  shows  that  the  slip  of  the  block  increases  both  ways  from 
its  centre,  as  had  been  predicted.  The  arc  for  u  =  o  is  the 
standard  with  which  the  other  curves  must  be  compared. 
CASE  II.  The  link  being  suspended  at  the  centre  of  Us 
chord. — Fig.  43  is  inserted  merely  for  the  purpose  of  show- 
ing an  imperfect  mode  of  suspension.  All  of  the  slip-curves 
are  bad,  and  at  no  point  is  there  any  cessation  of  the  slip. 

FIG.  44. 


CASE  III.  The  link  suspended  at  the  bottom. — Fig.  44  re- 
veals the  fact  that  the  slip  becomes  very  great  for  the  upper 
end  of  the  link — so  great  as  to  seriously  affect  the  distribution 
of  the  steam.  The  lower  half  of  the  link  only  can  be  relied 
upon  for  accurate  work. 

CASE  IV.  The  link  suspended  halfway  between  the  bottom 
and  centre. — Fig.  45  shows  a  better  average  result  than  any 
of  the  others,  and  is  undoubtedly  the  best  mode  of  hanging 


OF  THE  STEAM-ENGINE. 


209 


the  link  when  the  grade  u  =  -  £c  is  to  be  generally  used. 
Viewed  from  a  practical  point,  slip  is  of  great  importance, 
being  the  cause  of  the  wear  upon  links,  which  soon  unfits 


FIG.  45. 


them  for  accurate  work.  Great  pains  are  taken  to  reduce 
this  wear  by  case-hardening  the  links  or  using  steel  in  the 
place  of  wrought  iron. 

A  proper  mode  of  suspension  is  the  most  important  point 
to  be  attained  when  the  durability  of  the  link-motion  is  under 
consideration. 

18* 


TABLES. 


THE  following  four  tables,  .condensed  from  the  fourth 
section  of  Weisbach's  Mechanics  of  Engineering,  which  has 
formed  the  basis  of  this  work,  are  inserted  as  a  means  of 
ready  reference  for  ordinary  problems  in  the  strength  and 
elasticity  of  materials.  The  same  notation  as  that  used  by 
Weisbach  is  retained,  in  order  to  avoid  confusion  in  refer- 
ring to  his  work. 

211 


212  THE  RELATIVE  PROPORTIONS 


TABLE  V. 

(75)   Elasticity  and  Strength  of  Extension  and  Com- 
pression. (Arts.  201-214.) 

(Art.  204.)  To  find  increase  or  decrease  in  length  under  a  strain 
of  extension  or  compression. 

Where  the  weight  of  the  body  under  strain  is  not  considered  — 

X  =  the  amount  of  the  extension  in  inches. 
P=  the  weight  acting  in  pounds. 

Z  =  the  length  of  the  body  acted  upon  in  inches. 

F=  the  area  of  the  cross-section  in  square  inches. 
J£=the  modulus  of  elasticity  in  pounds  per  square  inch. 

A      Pl 
~  FE' 

Let  (7=  the  weight  of  the  body  under  strain. 

(Art.  207.)  Where  the  weight  of  the  body  under  the  strain  is  also 
taken  into  account,  %=-  —   * 


(Art.  205.)  To  find  the  proof-strength  of  a  body  to  be  submitted 
to  strain. 

T  =the  proof-strength  for  extension  per  square  inch  in  pounds. 
jrt  =  the  proof-strength  for  compression  per  square  inch  in  pounds. 
R  =  the  ultimate  strength  for  extension  per  square  inch  in  pounds. 
J£i  =  the  ultimate  strength  for  compression  per  square  inch  in  pounds. 

For  a  pull,  P=  FT.    For  a  thrust,  Pl  =  FTV 

To  find  the  ultimate  strength  of  a  body  to  be  submitted  to  strain. 

To  tear  the  body  asunder,  P=  FK.    To  crush  it,  Pl  = 


OF 


STEAM-ENGINE. 


213 


S-   B 
>   §* 

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gg 

E  5 

I 

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Pfy 
5 

B 

L.         » 

s  I 
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I  & 


15 

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M 

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Is 
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R 

f 

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2-  "a 


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i: 

g£ 

£§ 
2?  §. 

I 

1 


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11 

p'S"5'« 


r 


^00 


.- 

3W& 
Art.  217 


d    I  *  B 

IIHII: 

«    11  fa 


|t.| 

iP 

•y  ^  s 

*  ^ 

M~'S.  ^•'*s 

US.  p*_ 


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•K 

H 

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I? 


gs 


H 

l&sg 


nl 


IB  » 

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S"  f 

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p*  to 


214 


THE  RELATIVE  PROPORTIONS 


TABLE  VII. 

(77)    The  Elasticity  and  Strength  of  Torsion.   (Arts. 
262-265.) 

Notation. — P=  the  load  in  pounds,  a  =  the  lever-arm  of  the  load 
in  inches.  T=  the  modulus  of  proof-strength  for  shearing  in  pounds 
per  square  inch.  TF— the  measure  of  moment  of  torsion.  e  =  the 
greatest  distance  in  inches  of  any  element  of  the  cross-section  from 
the  neutral  axis. 

Z=the  length  in  inches  submitted  to  stress. 
d=the  diameter  of  round  shafts  in  inches. 
6  =  the  length  of  one  side  of  a  square  shaft  in  inches. 
a°  =  the  angle  through  which  the  body  is  twisted  in  degrees. 


Form  of  body. 

Proof-strength. 

Angle  of  torsion. 

For  any  form  of  cross-  \ 

TW 

section  j 

JL=           . 

OM5 

See  Art.  264. 

Cast  iron  (Art.  263). 

o°-  0.0002053^. 

rPT 

d* 

For  a  solid  round  shaft... 

P=  0.1963—.- 
a 

See  Art.  264. 

Wro't  iron  (Art.  263). 
p  i  » 

a°  =  0.0000648^-. 

Cast  iron  (Art.  263). 

PnJ 

3 

a°  =  0.0001211^. 

For  a  solid  square  shaft.. 

P=  0.2357—  .- 
a 

See  Art.  264. 

Wro't  iron  (Art.  263). 
pi 

a°  =  0.0000382^-. 

OF  THE  STEAM-ENGINE. 


215 


TABLE  VIII. 

(78)   The  Proof-Strength  of  Long  Columns.    (Arts. 
265  to  270.) 

/  =  the  length  of  the  column  in  inches. 
d  =  the  diameter  of  the  column  in  inches. 

When  the  length  of  columns  is  so  increased  as  to  cause  rupture  by 
first  bending  and  then  breaking  across  (buckling). 
The  following  formulae  will  apply  approximately : 


Method  of  adjustment. 

Force  necessary  to 
rupture  by  buck- 
ling. 

Remarks. 

Column  fast  at  the  lower  end,  ") 
load  applied  at  the  upper  1 
end,  which  is  free  to  move  [ 
sideways  ..  J 

p-(£)V* 

See  Art  265 

W  and  E  are  the 
same  as  in  the 
case  of  flexure. 

Column  not  fixed  at  either  ~\ 
end,  but  neither  end  free  [• 
to  move  sideways  J 

4P. 

See  Art  266 

Column  fixed  at  both  ends,  ~| 
and  not  free  to  move  side-  >• 

16P. 
See  Art.  266. 

According  to  Hodg- 
kinson's  experi- 
ments, we  have 

only  12P. 

(Art.  266.)  For  a  solid  cylindrical  pillar  not  fixed  at  either  end, 
but  neither  end  free  to  move  sideways,  whose  diameter  is  d  and  whose 
length  is  I,  we  must  take  the  formulae  for  buckling,  in  preference  to 
that  for  compression. 

In  the  case  of  cast  iron  when          j  =  9  or  a  larger  number. 
In  the  case  of  wrought  iron  when  -  =  22  or  a  larger  number. 


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